intro physics

Review Problemsfor

Introductory Physics 1

June 12, 2014

Robert G. Brown, Instructor

Duke University Physics Department

Durham, NC 27708-0305

[email protected]

Copyright Notice

This document is Copyright Robert G. Brown 2011. The specific date of lastmodification is determinable by examining the original document sources.

Contents

1 Preface 3

2 Short Math Review Problems 5

3 Essential Laws, Theorems, and Principles 11

4 Problem Solving 19

5 Newton’s Laws 21

5.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.1.1 Multiple Choice . . . . . . . . . . . . . . . . . . . . . . . . 22

5.1.2 Short Answer . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.1.3 Long Problems . . . . . . . . . . . . . . . . . . . . . . . . 27

5.2 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.2.1 Multiple Choice . . . . . . . . . . . . . . . . . . . . . . . . 31

5.2.2 Ranking/Scaling . . . . . . . . . . . . . . . . . . . . . . . 52

5.2.3 Short Answer . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.2.4 Long Problems . . . . . . . . . . . . . . . . . . . . . . . . 62

5.3 Circular Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.3.1 Multiple Choice . . . . . . . . . . . . . . . . . . . . . . . . 93

5.3.2 Long Problems . . . . . . . . . . . . . . . . . . . . . . . . 96

6 Work and Energy 103

i

ii CONTENTS

6.1 Work and Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . 104

6.1.1 Multiple Choice . . . . . . . . . . . . . . . . . . . . . . . . 104

6.1.2 Ranking/Scaling . . . . . . . . . . . . . . . . . . . . . . . 107

6.1.3 Short Answer . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.1.4 Long Problems . . . . . . . . . . . . . . . . . . . . . . . . 114

6.2 Work and Mechanical Energy . . . . . . . . . . . . . . . . . . . . 119

6.2.1 Multiple Choice . . . . . . . . . . . . . . . . . . . . . . . . 119

6.2.2 Ranking/Scaling . . . . . . . . . . . . . . . . . . . . . . . 121

6.2.3 Short Answer . . . . . . . . . . . . . . . . . . . . . . . . . 122

6.2.4 Long Problems . . . . . . . . . . . . . . . . . . . . . . . . 130

6.3 Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

6.3.1 Multiple Choice . . . . . . . . . . . . . . . . . . . . . . . . 150

6.3.2 Long Problems . . . . . . . . . . . . . . . . . . . . . . . . 151

7 Center of Mass and Momentum 153

7.1 Center of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

7.1.1 Multiple Choice . . . . . . . . . . . . . . . . . . . . . . . . 154

7.1.2 Short Answer . . . . . . . . . . . . . . . . . . . . . . . . . 160

7.1.3 Long Problems . . . . . . . . . . . . . . . . . . . . . . . . 161

7.2 Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

7.2.1 Multiple Choice . . . . . . . . . . . . . . . . . . . . . . . . 170

7.2.2 Short Answer . . . . . . . . . . . . . . . . . . . . . . . . . 179

7.2.3 Long Problems . . . . . . . . . . . . . . . . . . . . . . . . 184

8 One Dimensional Rotation and Torque 201

8.1 Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

8.1.1 Multiple Choice . . . . . . . . . . . . . . . . . . . . . . . . 202

8.1.2 Ranking/Scaling . . . . . . . . . . . . . . . . . . . . . . . 206

8.1.3 Short Answer . . . . . . . . . . . . . . . . . . . . . . . . . 208

8.1.4 Long Problems . . . . . . . . . . . . . . . . . . . . . . . . 211

CONTENTS iii

8.2 Moment of Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . 240

8.2.1 Ranking/Scaling . . . . . . . . . . . . . . . . . . . . . . . 240

8.2.2 Short Answer . . . . . . . . . . . . . . . . . . . . . . . . . 243

8.2.3 Long Problems . . . . . . . . . . . . . . . . . . . . . . . . 247

9 Vector Torque and Angular Momentum 253

9.1 Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . 254

9.1.1 Multiple Choice . . . . . . . . . . . . . . . . . . . . . . . . 254

9.1.2 Short Answer . . . . . . . . . . . . . . . . . . . . . . . . . 257

9.1.3 Long Problems . . . . . . . . . . . . . . . . . . . . . . . . 259

9.2 Vector Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

9.2.1 Multiple Choice . . . . . . . . . . . . . . . . . . . . . . . . 270

9.2.2 Short Answer . . . . . . . . . . . . . . . . . . . . . . . . . 271

9.2.3 Long Problems . . . . . . . . . . . . . . . . . . . . . . . . 279

10 Statics 287

10.1 Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288

10.1.1 Multiple Choice . . . . . . . . . . . . . . . . . . . . . . . . 288

10.1.2 Ranking/Scaling . . . . . . . . . . . . . . . . . . . . . . . 301

10.1.3 Short Answer . . . . . . . . . . . . . . . . . . . . . . . . . 307

10.1.4 Long Problems . . . . . . . . . . . . . . . . . . . . . . . . 314

11 Fluids 343

11.1 Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344

11.1.1 Multiple Choice . . . . . . . . . . . . . . . . . . . . . . . . 344

11.1.2 Ranking/Scaling . . . . . . . . . . . . . . . . . . . . . . . 355

11.1.3 Short Answer . . . . . . . . . . . . . . . . . . . . . . . . . 361

11.1.4 Long Problems . . . . . . . . . . . . . . . . . . . . . . . . 374

12 Oscillations 403

12.1 Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404

CONTENTS 1

12.1.1 Multiple Choice . . . . . . . . . . . . . . . . . . . . . . . . 404

12.1.2 Ranking/Scaling . . . . . . . . . . . . . . . . . . . . . . . 409

12.1.3 Short Answer . . . . . . . . . . . . . . . . . . . . . . . . . 422

12.1.4 Long Problems . . . . . . . . . . . . . . . . . . . . . . . . 430

13 Waves 463

13.1 Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464

13.1.1 Multiple Choice . . . . . . . . . . . . . . . . . . . . . . . . 464

13.1.2 Ranking/Scaling . . . . . . . . . . . . . . . . . . . . . . . 468

13.1.3 Short Answer . . . . . . . . . . . . . . . . . . . . . . . . . 469

13.1.4 Long Problems . . . . . . . . . . . . . . . . . . . . . . . . 478

14 Sound 487

14.1 Sound Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488

14.1.1 Multiple Choice . . . . . . . . . . . . . . . . . . . . . . . . 488

14.1.2 Ranking/Scaling . . . . . . . . . . . . . . . . . . . . . . . 495

14.1.3 Short Answer . . . . . . . . . . . . . . . . . . . . . . . . . 496

14.1.4 Long Problems . . . . . . . . . . . . . . . . . . . . . . . . 505

15 Gravity 515

15.1 Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516

15.1.1 Multiple Choice . . . . . . . . . . . . . . . . . . . . . . . . 516

15.1.2 Ranking/Scaling . . . . . . . . . . . . . . . . . . . . . . . 521

15.1.3 Short Answer . . . . . . . . . . . . . . . . . . . . . . . . . 522

15.1.4 Long Problems . . . . . . . . . . . . . . . . . . . . . . . . 529

16 Leftovers 547

16.1 Leftover Short Problems . . . . . . . . . . . . . . . . . . . . . . . 547

2 CONTENTS

Chapter 1

Preface

The problems in this review guide are provided as is without any guaranteeof being correct! That’s not to suggest that they are all broken – on thecontrary, most of them are well-tested and have been used as homework, quizand exam problems for decades if not centuries. It is to suggest that they havetypos in them, errors of other sorts, bad figures, and one or two of them arereally too difficult for this course but haven’t been sorted out or altered to makethem doable.

Leaving these in just adds to the fun. Physics problems are not all cut anddried; physics itself isn’t. One thing you should be building up as you work isan appreciation for what is easy, what is difficult, what is correct and what isincorrect. If you find an error and bring it to my attention, I’ll do my best tocorrect it, of course, but in the meantime, be warned!

A few of the problems have rather detailed solutions (due to Prof. RonenPlesser and myself), provided as examples of how a really good solution mightdevelop, with considerable annotation. However, most problems do not haveincluded solutions and never will have. I am actually philosophically opposedto providing students with solutions that they are then immediately tempted tomemorize. This guide is provided so that you can learn to solve problems andwork sufficiently carefully that they can trust the solutions.

Students invariably then ask: But how are we to know if we’ve solved theproblems correctly?”

The answer is simple. The same way you would in the real world! Work onthem in groups and check your algebra, your approach, and your answers againstone another’s. Build a consensus. Solve them with mentoring (course TAs,professors, former students, tutors all are happy to help you). Find answersthrough research on the web or in the literature.

3

4 CHAPTER 1. PREFACE

To be honest, almost any of the ways that involve hard work on your part aregood ways to learn to solve physics problems. The only bad way to (try to) learnis to have the material all laid out, cut and dried, so that you don’t have tostruggle to learn, so that you don’t have to work hard and thereby permanently

imprint the knowledge on your brain as you go. Physics requires engagementand investment of time and energy like no subject you have ever taken, if onlybecause it is one of the most difficult subjects you’ve ever tried to learn (at thesame time it is remarkably simple, paradoxically enough).

In any event, to use this guide most effectively, first skim through the wholething to see what is there, then start in at the beginning and work through it,again and again, reviewing repeatedly all of the problems and material you’vecovered so far as you go on to what you are working on currently in class and onthe homework and for the upcoming exam(s). Don’t be afraid to solve problemsmore than once, or even more than three or four times.

And work in groups! Seriously! With pizza and beer...

Chapter 2

Short Math ReviewProblems

The problems below are a diagnostic for what you are likely to need in orderto work physics problems. There aren’t really enough of them to constitutepractice”, but if you have difficulty with any of them, you should probably finda math review (there is usually one in almost any introductory physics text andthere are a number available online) and work through it.

Weakness in geometry, trigonometry, algebra, calculus, solving simultaneousequations, or general visualization and graphing will all negatively impact yourphysics performance and, if uncorrected, your grade.

5

6 CHAPTER 2. SHORT MATH REVIEW PROBLEMS

Short Problem 1.

problems/short-math-binomial-expansion.tex

Write down the binomial expansion for the following expressions, given theconditions indicated. FYI, the binomial expansion is:

(1 + x)n = 1 + nx +n(n − 1)

2!x2 +

n(n − 1)(n − 2)

3!x3 + . . .

where x can be positive or negative and where n is any real number and onlyconverges if |x| < 1. Write at least the first three non-zero terms in the expan-sion:

a) For x > a:1

(x + a)2

b) For x > a:1

(x + a)3/2

c) For x > a:(x + a)1/2

d) For x > a:1

(x + a)1/2− 1

(x − a)1/2

e) For r > a:1

(r2 + a2 − 2ar cos(θ))1/2

7

Short Problem 2.

problems/short-math-elliptical-trajectory.tex

The position of a particle as a function of time is given by:

~x(t) = x0 cos(ωt)x + y0 sin(ωt)y

where x0 > y0.

a) What is ~v(t) for this particle?

b) What is ~a(t) for this particle?

c) Draw a generic plot of the trajectory function for the particle. What kindof shape is this? In what direction/sense is the particle moving (indicatewith arrow on trajectory)?

d) Draw separate plots of x(t) and y(t) on the same axes.


Short Problem 3.

problems/short-math-simple-series.tex

Evaluate the first three nonzero terms for the Taylor Series for the followingexpressions. Recall that the radius of convergence for the binomial expansion(another name for the first taylor series in the list below) is |x| < 1 – this givesyou two ways to consider the expansions of the form (x + a)n.

a) Expand about x = 0:

(1 + x)−2 ≈

b) Expand about x = 0:

ex ≈

c) For x > a (expand about x or use the binomial expansion after factoring):

(x + a)−2 ≈

d) Estimate 0.91/4 to within 1% without a calculator, if you can. Explainyour reasoning.

9

Short Problem 4.

problems/short-math-sum-two-vectors-1.tex

Suppose vector ~A = −4x + 6y and vector ~B = 9x + 6y. Then the vector~C = ~A + ~B:

a) is in the first quadrant (x+,y+) and has magnitude 17.

b) is in the fourth quadrant (x+,y-) and has magnitude 12.

c) is in the first quadrant (x+,y+) and has magnitude 13.

d) is in the second quadrant (x-,y+) and has magnitude 17.

e) is in the third quadrant (x-,y-) and has magnitude 13.


Short Problem 5.

problems/short-math-taylor-series.tex

Evaluate the first three nonzero terms for the Taylor series for the followingexpressions. Expand about the indicated point:

a) Expand about x = 0:(1 + x)n ≈

b) Expand about x = 0:sin(x) ≈

c) Expand about x = 0:cos(x) ≈

d) Expand about x = 0:ex ≈

e) Expand about x = 0 (note: i2 = −1):

eix ≈

Verify that the expansions of both sides of the following expression match:

eiθ = cos(θ) + i sin(θ)

Chapter 3

Essential Laws, Theorems,and Principles

The questions below guide you through basic physical laws and concepts. Theyare the stuff that one way or another you should know” going into any examor quiz following the lecture in which they are covered. Note that there aren’treally all that many of them, and a lot of them are actually easily derived fromthe most important ones.

There is absolutely no point in memorizing solutions to all of the problems inthis guide. In fact, for all but truly prodigious memories, memorizing them allwould be impossible (presuming that one could work out all of the solutions intoan even larger book to memorize!). However, every student should memorize,internalize, learn, know the principles, laws, and and theorems covered in thissection (and perhaps a few that haven’t yet been added). These are things uponwhich all the rest of the solutions are based.

11

12 CHAPTER 3. ESSENTIAL LAWS, THEOREMS, AND PRINCIPLES

Short Problem 1.

problems/true-facts-angular-momentum-conservation.tex

When is the angular momentum of a system conserved?

Short Problem 2.

problems/true-facts-archimedes-principle.tex

What is Archimedes’ Principle? (Equation with associated diagram or clear and

correct statement in words.)

Short Problem 3.

problems/true-facts-bernoullis-equation.tex

What is Bernoulli’s equation? What does it describe? Draw a small picture toillustrate.

Short Problem 4.

problems/true-facts-coefficient-of-performance.tex

How is the coefficient of performance of a refrigerator defined? Draw a smalldiagram that schematically indicates the flow of heat and work between reser-voirs.

Short Problem 5.

problems/true-facts-conditions-static-equilibrium.tex

What are the two conditions for a rigid object to be in static equilibrium?

13

Condition 1:

Condition 1:

Short Problem 6.

problems/true-facts-coriolis-force.tex

What “force” makes hurricanes spin counterclockwise in the northern hemi-sphere and clockwise in the southern hemisphere?

Short Problem 7.

problems/true-facts-definition-of-decibel.tex

One measures sound intensity in decibels. What is a decibel? (Equation, please,and define and give value of all constants.)

Short Problem 8.

problems/true-facts-doppler-shift-moving-source.tex

What is the equation for the Doppler shift, specifically for the frequency f ′

heard by a stationary observer when a source emitting waves with speed u atfrequence f0 is approaching at speed us?

Short Problem 9.

problems/true-facts-equipartition-theorem.tex

What is the Equipartition Theorem?

Short Problem 10.

problems/true-facts-four-forces-of-nature.tex

Name the four fundamental forces of nature as we know them now.


a)

b)

c)

d)

Short Problem 11.

problems/true-facts-generalized-work-energy.tex

What is the Generalized Work-Mechanical-Energy Theorem? (Equation only.This is the one that differentiates between conservative and non-conservativeforces.)

Short Problem 12.

problems/true-facts-heat-capacity-monoatomic-gas.tex

What is the heat capacity at constant volume CV of N molecules of an idealmonoatomic gas? What is its heat capacity at constant pressure CP ?

Short Problem 13.

problems/true-facts-heat-engine-efficiency.tex

What is the algebraic definition of the efficiency of a heat engine? Draw asmall diagram that schematically indicates the flow of heat and work betweenreservoirs.

Short Problem 14.

problems/true-facts-inelastic-collision-conservation.tex

What is conserved (and what isn’t) in an inelastic collision?

Short Problem 15.

problems/true-facts-integral-definition-moment-of-inertia.tex

15

Write the integral definition of the moment of inertia of an object about aparticular axis of rotation. Draw a picture illustrating what “dm” is within theobject relative to the axis of rotation.

Short Problem 16.

problems/true-facts-kepler1.tex

What is Kepler’s First Law?

Short Problem 17.


What is Kepler’s Second Law and what physical principle does it correspondto?

Short Problem 18.


What is Kepler’s Third Law?

Short Problem 19.

problems/true-facts-momentum-conservation.tex

Under what condition(s) is the linear momentum of a system conserved?

Short Problem 20.

problems/true-facts-n1.tex

What is Newton’s First Law?

Short Problem 21.



What is Newton’s Second Law?

Short Problem 22.


What is Newton’s Third Law?

Short Problem 23.

problems/true-facts-newtons-law-gravitation.tex

What is Newton’s Law for Gravitation? Draw a picture showing the coordinatesused (for two pointlike masses at arbitrary positions), and indicate the value ofG in SI units.

Short Problem 24.

problems/true-facts-parallel-axis-theorem.tex

Write the parallel axis theorem for the moment of inertia of an object aroundan axis parallel to one through its center of mass. Draw a picture to go with it,if it helps.

Short Problem 25.

problems/true-facts-pascals-principle.tex

What is Pascal’s principle? A small picture would help.

Short Problem 26.

problems/true-facts-perpendicular-axis-theorem.tex

Write the perpendicular axis theorem for a mass distributed in the x− y plane.Draw a picture to go with it, if it helps.

Short Problem 27.

17

problems/true-facts-toricellis-law.tex

What is Toricelli’s Law (for fluid flow) and what is the condition required for itto be approximately true?

Short Problem 28.

problems/true-facts-venturi-effect.tex

What is the Venturi Effect?

Short Problem 29.

problems/true-facts-wave-equation-string.tex

Write the wave equation (the differential equation) for waves on a string withtension T and mass density µ. Identify all parts.

Short Problem 30.

problems/true-facts-work-energy.tex

What is the Work-Kinetic Energy Theorem?

Short Problem 31.

problems/true-facts-youngs-modulus.tex

What is the definition of Young’s modulus Y ? Draw a picture illustrating thephysical situation it describes and define all terms used in terms of the picture.

Chapter 4

Problem Solving

The following problems are, at last, the meat of the matter: serious, moder-ately to extremely difficult physics problems. An A” student would be able toconstruct beautiful solutions, or almost all, of these problems.

Note well the phrase beautiful solutions”. In no case is the answer” to theseproblems an equation, or a number (or set of equations or numbers). It is aprocess. Skillful physics involves a systematic progression that involves:

• Visualization and conceptualization. What’s going on? What will hap-pen?

• Drawing figures and graphs and pictures to help with the process of de-termining what physics principles to use and how to use them. The papershould be an extension of your brain, helping you associate coordinatesand quantities with the problem and working out a solution strategy. Forexample: drawing a free body diagram” in a problem where there arevarious forces acting on various bodies in various directions will usuallyhelp you break a large, complex problem into much smaller and moremanageable pieces.

• Identifying (on the basis of these first two steps) the physical principles touse in solving the problem. These are almost invariably things from theLaws, Theorems and Principles chapter above, and with practice, you willget to where you can easily identify a Newtons Second Law” problem (orpart of a problem) or an Energy Conservation” (part of) a problem.

• Once these principles are identified (and identifying them by name is agood practice, especially at first!) one can proceed to formulate the solu-tion. Often this involves translating your figures into equations using thelaws and principles, for example creating a free body diagram and trans-

19

20 CHAPTER 4. PROBLEM SOLVING

lating it into Newton’s Second Law for each mass and coordinate directionseparately.

• At this point, believe it or not, the hard part is usually done (and most ofthe credit for the problem is already secured). What’s left is using alge-

bra and other mathematical techniques (e.g. trigonometry, differentiation,integration, solution of simultaneous equations that combine the resultsfrom different laws or principles into a single answer) to obtain a com-

pletely algebraic (symbolic) expression or set of expressions that answerthe question(s).

• At this point you should check your units! One of several good reasonsto solve the problem algebraically is that all the symbols one uses carryimplicit units, so usually it is a simple matter to check whether or notyour answer has the right ones. If it does, that’s good! It means youprobably didn’t make any trivial algebra mistakes like dividing instead ofmultiplying, as that sort of thing would have led to the wrong units.

Remember, an answer with the right units may be wrong, but it’s notcrazy and will probably get lots of credit if the reasoning process is clear.On the other hand, an answer with the wrong units isn’t just mistaken,its crazy mistaken, impossible, silly. Even if you can’t see your error, ifyou check your answer and get the wrong units say so; your instructor canthen give you a few points for being diligent and checking and knowingthat you are wrong, and can usually quickly help you find your mistakeand permanently correct it.

• Finally, at the very end, substitute any numbers given for the algebraicsymbols, do the arithmetic, and determine the final numerical answers.

Most of the problems below won’t have any numbers in them at all to emphasizehow unimportant this last step is in learning physics! Sure, you should learnto be careful in your doing of arithmetic, but anybody can (with practice)learn to punch numbers into a calculator or enter them into a computer thatwill do all of the arithmetic flawlessly no matter what. It is the process ofdetermining how to punch those numbers or program the computer to evaluatea correct formula that is what physics is all about. Indeed, with skill andpractice (especially practice at estimation and conceptual problem solving) youwill usually be able to at least approximate an answer and fully understandwhat is going on and what will happen even without doing any arithmetic atall, or doing only arithmetic you can do in your head.

As with all things, practice makes perfect, wax on, wax off, and the more fun

you have while doing, the more you will learn. Work in groups, with friends,over pizza and beer. Learning physics should not be punishment, it should bea pleasure. And the ultimate reward is seeing the entire world around you withdifferent eyes...

Chapter 5

Newton’s Laws

The following problems include both kinematics” problems that can be solvedusing nothing but mathematics with units (no real physical principle but thenotion of acceleration, for example) and true dynamics problems – ones basedon Newton’s Second Law plus various force laws, used to find the acceleration(s)of various masses and, the differential equation(s) of motion of those masses,and from those differential equations (by integration or later in the course, bymore complicated methods of solution) the velocity and the position.

Some problems are very short and can be answered on the basis of physicsconcepts with little or no computation. Others must be solved algebraicallywith some real effort, usually using multiple physics concepts and several math-ematical skills synthesized together. A few problems are presented and thencompletely solved to illustrate good solution methodology.

The main thing to remember is – be sure to connect the basic physicalprinciples involved to each problem as you solve them. Write them downseparately, in English as well as equation form (if there is one). Well over halfof the difficulty of solving physics problems on exams comes from being cluelessconcerning what the problem is all about. Typically, anywhere from 1/3 to 1/2the credit for a problem can be obtained by simply getting the basic physicsright even if you screw up the algebra afterwards or have no idea how to proceedfrom the concept to the solution.

But you will find, especially with practice, that once you do learn to identify theright physics to go with a problem you can usually make it through the algebraand other math to get a creditable solution, presuming only that you honestlyhave met the mathematical prerequisites for this course (or have beefed up yourmath skills along the way).

Good luck!

21

22 CHAPTER 5. NEWTON’S LAWS

5.1 Kinematics

5.1.1 Multiple Choice

Problem 1.

problems/kinematics-mc-shooting-the-monkey.tex

A zookeeper wants to shoot a monkey sitting on a branch a height H abovethe gun muzzle in a tree a horizontal distance R away with a tranquilizer gun.Where must the zookeeper aim the gun in order to hit the monkey if the monkeyfalls asleep in the tree after being shot (that is, does not drop from the tree atthe instant he fires)? (Neglect air resistance, justify your answer with a sketchor some work.)

a) Straight at the monkey.

b) Slightly below the monkey.

c) Slightly above the monkey.

d) Cannot tell without knowing the mass of the monkey and the dart.

5.1. KINEMATICS 23

Problem 2.

problems/kinematics-mc-velocity-acceleration-2D-1.tex

C

B

A

v

+y

+x

A small ball of mass m is thrown so that it follows the parabolic trajectoryshown. Neglect drag forces. Circle the true statement(s) (there can be morethan one) below:

a) The minimum speed occurs at point A.

b) The maximum speed occurs at point C.

c) The acceleration is larger at point B than it is at point C.

d) The acceleration is the same at points A and C.

e) The speed is the same at points A and C.


Problem 3.

problems/kinematics-mc-velocity-acceleration-2D-2.tex

C

B

A

v

+y

+x

A small ball of mass m is thrown so that it follows the parabolic trajectoryshown. Neglect drag forces. Circle the true statement(s) (there can be morethan one) below:

a) The speed is the same at points A and C.

b) The maximum speed occurs at point C.

c) The minimum speed occurs at point A.

d) The acceleration is the same at points A and C.

e) The acceleration is larger at point B than it is at point C.

5.1. KINEMATICS 25

5.1.2 Short Answer

Problem 4.

problems/kinematics-sa-a2v2x.tex

-20

-15

-10

-5

0

5

10

15

20

0 2 4 6 8 10

x (m)

t (seconds)

-10

-5

0

5

10

0 2 4 6 8 10

v (m/sec)

t (seconds)

-10

-5

0

5

10

0 2 4 6 8 10a (m/sec2)

t (seconds)

On the figure above the acceleration a(t) for one-dimensional motion is plottedin the bottom panel. In the first and second panel sketch in approximate curvesthat represent x(t) and v(t) respectively. Your curves should at least qualita-

tively agree with the bottom figure. Assume that the particle in question startsfrom rest and is at the origin at time t = 0.


Problem 5.

problems/kinematics-sa-scaling-fall-times.tex

If a mass (near the surface of the Earth) that is dropped from rest from 5 meterstakes 1 second to reach the ground, roughly how high do you have to drop itfrom for it to take 3 seconds to reach the ground? (Neglect air resistance andshow work to justify your answer.)

5.1. KINEMATICS 27

5.1.3 Long Problems

Problem 6.

problems/kinematics-pr-stopping-before-a-bicycle.tex

vs bv

Sally is driving on a straight country road at night at a speed of vs = 20meters/second when she sees a bicyle without any lights loom ahead of her. Sheslams on her brakes when she is a distance D = 50 meters from the bike.

a) The bike is travelling at vb = 5 meters/second in the same direction thatSally is travelling. The maximum braking acceleration her tires can exertis a = −5 meters/second2. The biker is deaf and doesn’t hear her brakesor alter his speed in any way. Does she hit the bike?

b) Suppose that the bike is travelling towards her (on the wrong side of theroad). Does she hit the bike then?

Please solve the problems algebraically, and only at the end insert numbers(which one can do in one’s head, no calculator needed). Note that this is whyone should always ride a bike with traffic, and never against it. There simplyisn’t time to stop!


Problem 7.

problems/kinematics-pr-stopping-before-a-turtle.tex

A distance of D meters ahead of your car you see a box turtle crossing the road.Your car is traveling at a speed of v meters per second straight at the turtle(along the straight road).

a) What is the minimum acceleration you car must have in order to stopbefore hitting the turtle?

b) How long does it take to stop your car at this acceleration?

5.1. KINEMATICS 29

Problem 8.

problems/kinematics-pr-two-braking-cars.tex

v 2a 3v/2 a

D1 2

Two cars are driving down a straight country road when the driver of the firstcar (travelling at speed v1 = v) sees a turtle crossing the road in front of her.She quickly applies the brake, causing her car to slow down with a (negative)acceleration a1 = 2a. The second car is a distance D behind her and is travellingat an initial speed v2 = 3v/2. Its driver immediately applies his brakes as well– assume at the same time as the driver of the first car – but his car is heavierand his tires are not so good and his car only slows down with a (negative)acceleration a2 = a.

a) Find the minimum value of D such that the cars do not collide.

b) Qualitatively graph x1(t) and x2(t) as functions of time on a reasonablescale.

c) Qualitatively graph v1(t) and v2(t) as functions of time on a reasonablescale with the same time axis.


Problem 9.

problems/kinematics-pr-two-falling-balls.tex

1 2

H

v0

R

When a trigger is pulled at time t = 0, a compressed spring simultaneouslydrops ball 1 and hits identical ball 2 so that it is shot out to the right as initialspeed v0 as shown. The two balls then independently fall a height H . Answerthe following questions, assuming that the balls fall only under the influence ofgravity. (Neglect drag forces, and express all answers in terms of the givens, inthis case H and v0 and (assumed) gravitational acceleration g.).

a) Which ball strikes the ground first (or do they strike at the same time)?Prove your answer by finding the time that each ball hits the ground.

b) Which ball is travelling faster when it hits the ground (or do they hitat the same speed)? Prove your answer by finding an expression for thespeed each ball has when it hits the ground.

c) Find an expression for R, the horizontal distance ball 2 travels beforehitting the ground.

5.2. DYNAMICS 31

5.2 Dynamics


Problem 10.

problems/force-mc-acceleration-two-masses-1.tex

H

m = 2 kg1 2m = 4 kg

+x

+y

A mass m1 = 2 kg and a mass of m2 = 4 kg are both dropped from rest fromthe same height at the same time. Circle the true statement(s) (there can bemore than one) below. Neglect drag forces.

a) While the two masses are falling, the force acting on m1 and the forceacting on m2 are equal in magnitude.

b) While the two masses are falling, the acceleration of m1 and the acceler-ation of m2 are equal in magnitude.

c) Mass m2 will strike the ground first.

d) Mass m1 will strike the ground first.

e) The two masses will strike the ground at the same time.


Problem 11.

problems/force-mc-acceleration-two-masses-2.tex

H

m = 2 kg1 2m = 4 kg

+x

+y

A mass m1 = 2 kg and a mass of m2 = 4 kg are both dropped from rest fromthe same height at the same time. Circle the true statement(s) (there can bemore than one) below. Neglect drag forces.

a) The two masses will strike the ground at the same time.

b) Mass m1 will strike the ground first.

c) Mass m2 will strike the ground first.

d) While the two masses are falling, the force acting on m1 and the forceacting on m2 are equal in magnitude.

e) While the two masses are falling, the acceleration of m1 and the acceler-ation of m2 are equal in magnitude.

5.2. DYNAMICS 33

Problem 12.

problems/force-mc-battleship.tex

A battleship simultaneously fires two shells at enemy ships along the trajec-tories shown. One ship (A) is close by; the other ship (B) is far away. Whichship gets hit first?

A B


Problem 13.

problems/force-mc-block-on-paper.tex

m

A block of mass m is resting on a long piece of smooth paper. The block hascoefficient of static and kinetic friction µs, µk with the paper, respectively. Youjerk the paper horizontally so it slides out from under the block quickly inthe direction indicated by the arrow without sticking. Which of the followingstatements about the force acting on and acceleration of the block are true

a) F = µsmg, a = µsg, both to the right.

b) F = µkmg to the right, a = µkg to the left.

c) F = µkmg, a = µkg both to the left.

d) F = µkmg to the left, a = µkg to the right.

e) F = µkmg, a = µkg both to the right.

f) None of the above.

5.2. DYNAMICS 35

Problem 14.

problems/force-mc-cannonball-timed-trajectories.tex

a b

Two cannons fire projectiles into the air along the trajectories shown. Neglectthe drag force of the air.

a) Cannonball a is in the air longer.

b) Cannonball b is in the air longer.

c) Cannonballs a and b are in the air the same amount of time.

d) We cannot tell which is in the air longer without more information thanis given in the picture.


Problem 15.

problems/force-mc-coriolis-dropped-mass-at-equator.tex

A dense mass m is dropped “from rest” from a high tower built at the equator.As the mass falls, it to a person standing on the ground appears to be deflectedas it falls to the:

a) East.

b) West.

c) North.

d) South.

e) Cannot tell from the information given.

5.2. DYNAMICS 37

Problem 16.

problems/force-mc-coriolis.tex

The Earth is a rotating sphere, and hence is not really an inertial reference frame.Select the true answers from the following list for the apparent behavior of e.g.naval projectiles or freely falling objects:

a) A naval projectile fired due North in the northern hemisphere will be(apparently) deflected East (spinward).

b) A naval projectile fired due South in the northern hemisphere will be(apparently) deflected East (spinward).

c) A bomb dropped from a helicopter hovering over a fixed point on thesurface in the northern hemisphere will be (apparently) deflected West(antispinward).

d) A bomb dropped from a helicopter hovering over a fixed point on thesurface in the northern hemisphere will be (apparently) deflected East(spinward).

e) An object placed at (apparent) “rest” on the surface of the Earth in theNorthern hemisphere experiences an (apparent) force to the North.

f) An object placed at (apparent) “rest” on the surface of the Earth in theNorthern hemisphere experiences an (apparent) force to the South.

g) The true weight of an object measured with a spring balance in a labora-tory on the equator is a bit larger than the measured weight.

h) The true weight of an object measured with a spring balance in a labora-tory on the equator is a bit smaller than the measured weight.


Problem 17.

problems/force-mc-falling-time-mars.tex

Surface gravity on Mars is roughly 1/3 that of the Earth. Suppose you drop arock (initially at rest) from a height Hm on Mars and it takes a time tg to hitthe ground. From what height He do you need to drop the mass on Earth sothat it hits the ground in the same amount of time?

a) He =√

3Hm

b) He = 3Hm

c) He = 9Hm

d) He = Hm/√

3

e) He = Hm/3

5.2. DYNAMICS 39

Problem 18.

problems/force-mc-falling-time-moon.tex

Surface gravity on the Moon is roughly 1/6 that of the Earth, and surface gravityon Mars is roughly 1/3 that of the Earth. Suppose you drop a rock (initially atrest) from a height Hmoon on the Moon and it takes a time t0 to hit the ground.From what height Hmars do you need to drop the mass on Mars so that it hitsthe ground in the same amount of time?

a) Hmars =√

2Hmoon

b) Hmars =√

12Hmoon

c) Hmars = 6Hmoon

d) Hmars = 3Hmoon

e) Hmars = 2Hmoon


Problem 19.

problems/force-mc-inertial-reference-frames.tex

Newton’s Second Law states that ~F tot = m~a where ~F tot is the total forceexerted on a given mass m by actual forces of nature or force rules that idealizeactual forces of nature (such as Hooke’s Law, normal forces, tension in a string),but only if one defines ~a in an inertial reference frame.

Select the best (most complete and accurate) explanation for the inertial refer-ence frame requirement below:

a) It is too difficult to solve for the acceleration in a non-inertial referenceframe.

b) Newton’s Third Law is only true in inertial reference frames, and is some-times needed to solve problems.

c) The Earth’s surface is an inertial reference frame, so we use it as the basisfor physics everywhere else.

d) The acceleration, mass and forces of nature are all the same in differentinertial reference frames so all of Newton’s Laws are valid there.

e) Because the inertia/mass of an object cannot be measured in a non-inertialreference frame, Newton’s Second Law doesn’t hold there.

5.2. DYNAMICS 41

Problem 20.

problems/force-mc-N3-pick-list-1.tex

The following sentences each describe two specific forces exerted by objects ina physical situation. Circle the letter of the sentences where those two forcesform a Newton’s Third Law force pair. More than one sentence or nosentences at all in the list may describe a Newton’s Third Law force pair.

a) In an evenly matched tug of war (where the rope does not move); teamone pulls the rope to the left with some force and team two pulls the ropeto the right with an equal magnitude force in the opposite direction.

b) Gravity pulls me down; the normal force exerted by a scale I’m standingon pushes me up with an equal magnitude force in the opposite direction.

c) The air surrounding a helium balloon pushes it up with a buoyant force;the balloon pushes the air down with an equal magnitude force in theopposite direction.


Problem 21.

problems/force-mc-N3-pick-list.tex

Identify the Newton’s Third Law pairs from the following list of forces (morethan one could be right):

a) The Earth pulls me down with gravity, the normal force exerted by theground pushes me up.

b) I pull hard on a rope in a game of tug-of-war, the rope pulls back hard onme.

c) A table exerts a force upwards on a book sitting on it; the book exerts aforce downward on the table.

d) I pull up on a fishing rod trying to land a big fish; the fish pulls down onthe fishing rod to get away.

e) My car is stuck in the mud. I push hard on the car to free it, but the carpushes back on me hard enough that I slip and fall on my face in the mire.

5.2. DYNAMICS 43

Problem 22.

problems/force-mc-pushing-two-blocks.tex

M mF

The figure shows two blocks of mass M and m that are being pushed along ahorizontal frictionless surface by a force of magnitude F as shown. What is themagnitude of the force that the block of mass M exerts on the block of massm?

a) F

b) m FM

c) m F(M+m)

d) M F(M+m)


Problem 23.

problems/force-mc-sliding-block-and-tackle.tex

m

m

1

2

s kµ µ,

A block of mass m1 sits on a rough table. The coefficient of static and kineticfriction between the mass and the table are µs and µk, respectively. Anothermass m2 is suspended as indicated in the figure above (where the pulleys aremassless and the string is massless and unstretchable). What is the maximummass m2 for which the blocks remain at rest?

a) m2 = 2m1µk

b) m2 = m1µk/2

c) m2 = m1/µs

d) m2 = 2m1µs

e) m2 = m1µs/2

5.2. DYNAMICS 45

Problem 24.

problems/force-mc-terminal-velocity-vsq-2bs.tex

Two spherical objects, both with mass m, are falling freely under the influenceof gravity through air. The air exerts a drag force on the two spheres in theopposite direction to their motion with magnitude F1 = b1v

21 and F2 = b2v

22

respectively, with b2 = 2b1.

Suppose the terminal speed for object 1 is vt. Then the terminal speed of object2 is:

a) 2vt

b)√

2vt

c) vt

d)√

22 vt

e) vt/2


Problem 25.

problems/force-mc-terminal-velocity-vsq.tex

m

g (down)

v

air

In the figure above, a spherical mass m is falling freely under the influence ofgravity through air. The air exerts a drag force on the sphere in the oppositedirection to its motion of magnitude Fd = bv2 (where the drag coefficient b isdetermined by the shape of the object and its interaction with the air). After a(long) time, the falling mass approaches a constant terminal speed vt, where:

a) vt = Fd

b

b) vt = mgb

c) vt =(

mgb

)2

d) vt =√

mgb

e) vt =(

Fd

m

)

t

5.2. DYNAMICS 47

Problem 26.

problems/force-mc-terminal-velocity-v.tex

m

g (down)

v

air

In the figure above, a spherical mass m is falling freely under the influence ofgravity through air. The air exerts a drag force on the sphere in the oppositedirection to its motion of magnitude Fd = bv (where the drag coefficient b isdetermined by the shape of the object and its interaction with the air). After a(long) time, the falling mass approaches a constant terminal speed vt, where:

a) vt = Fd

b

b) vt = mgb

c) vt =(

mgb

)2

d) vt =√

mgb

e) vt =(

Fd

m

)

t


Problem 27.

problems/force-mc-two-constant-forces.tex

x

y

F = 2 mggravity

m

In the figure above, a rocket engine exerts a constant force ~F = 2mg x to theright on a freely falling mass initially at rest as shown (no drag or frictionalforces are present). The object:

a) Moves in a straight line with an acceleration of magnitude 3g.

b) Moves in a straight line with an acceleration of magnitude√

5g.

c) Moves in a parabolic trajectory with an acceleration of magnitude 3g.

d) Moves in a parabolic trajectory with an acceleration of magnitude√

5g.

e) We cannot determine the trajectory and/or the magnitude of the acceler-ation from the information given.

Sketch your best guess for the trajectory of the particle in on the figure aboveas a dashed line with an arrow.

5.2. DYNAMICS 49

Problem 28.

problems/force-mc-two-masses-falling-drag-1.tex

H

m = 2 kg1

+x

+y m = 4 kg2

A mass m1 = 2 kg and a mass of m2 = 4 kg have identical size, shape, andsurface characteristics, and are both dropped from rest from the same height Hat the same time. Air resistance (drag force) is present! Circle the truestatement(s) below (more than one may be present).

a) Initially, the acceleration of both masses is the same.

b) The 2 kg mass hits the ground first.

c) The 4 kg mass hits the ground first.

d) Both masses hit the ground at the same time.

e) Just before they hit, the acceleration of the heavier mass is greater.


Problem 29.

problems/force-mc-two-masses-falling-drag-2.tex

H

m = 2 kg1

+x

+y m = 4 kg2

A mass m1 = 2 kg and a mass of m2 = 4 kg have identical size, shape, andsurface characteristics, and are both dropped from rest from the same height Hat the same time. Air resistance (drag force) is present! Circle the truestatement(s) below (more than one may be present).

a) Initially, the acceleration of both masses is the same.

b) Just before they hit, the acceleration of the lighter mass is greater.

c) The 2 kg mass hits the ground first.

d) The 4 kg mass hits the ground first.

e) Both masses hit the ground at the same time.

5.2. DYNAMICS 51

Problem 30.

problems/force-mc-weight-in-elevator.tex

m

a

In the figure above, a person of mass m is standing on a scale in an elevator(near the Earth’s surface) that is accelerating upwards with acceleration a.

a) The scale shows the person’s “weight” to be mg.

b) The scale shows the person’s “weight” to be ma.

c) The scale shows the person’s “weight” to be m(g − a).

d) The scale shows the person’s “weight” to be m(g + a).

e) We cannot tell what the scale would show without more information.


5.2.2 Ranking/Scaling

Problem 31.

problems/force-ra-circular-motion-tension.tex

= 2m

= 1 m

= 2 m/sec

= 1 m/sec

= 2 kg

= 1 kg

BA

C D

In the four figures above, you are looking down on a mass sitting on a frictionlesstable being whirled on the end of a string. The mass, length of string, and speedof the mass in each figure are indicated in the key on the right.

Rank the tension in the string in each of the four figures above, from lowestto highest. Equality is a possibility. An example of a possible answer is thus:A < B = C < D.

5.2. DYNAMICS 53

Problem 32.

problems/force-ra-tension-between-two-unequal-blocks-friction.tex

a)

b)

Tb

Ta

F

Fm

m

M

M

In the figure above a block of mass m is connected to a block of mass M > m bya string. Both blocks sit on a smooth surface with a coefficient of kinetic frictionµk between either block and the surface. In figure a), a force of magnitude F(large enough to cause both blocks to slide) is exerted on block M to pull thesystem to the right. In figure b), a force of (the same) magnitude F is exertedon block m to pull the system to the left.

Circle the true statement:

a) The tension Ta > Tb.

b) The tension Ta < Tb.

c) The tension Ta = Tb.

d) There is not enough information to determine the relative tension in thetwo cases.


Problem 33.

problems/force-ra-tension-between-two-unequal-blocks.tex

a)

b)

Tb

Ta

F

Fm

m

M

M

In the figure above a block of mass m is connected to a block of mass M > mby a string. Both blocks sit on a frictionless floor. In a), a force of magnitudeF is exerted on block M to pull the system to the right. In b), a force of (thesame) magnitude F is exerted on block m to pull the system to the left. Circlethe true statement:

a) The tension Ta > Tb.

b) The tension Ta < Tb.

c) The tension Ta = Tb.

d) There is not enough information to determine the relative tension in thetwo cases.

5.2. DYNAMICS 55

5.2.3 Short Answer

Problem 34.

problems/force-sa-block-on-paper.tex

mµk

A block of mass m is resting on a long piece of smooth paper. The block has acoefficient of kinetic friction µk with the paper. You pull the paper horizontallyout from under the block quickly in the direction indicated by the arrow.

a) Draw the direction of the frictional force acting on the block.

b) What is the magnitude and direction of the acceleration of the block?


Problem 35.

problems/force-sa-falling-time-earth-moon.tex

Gravity on the surface of the moon is weaker than it is on the surface of theearth: gmoon ≈ gearth/6. If a mass dropped from 5 meters above the earth takes1 second to reach the surface of the earth, how long does it take a mass droppedfrom the same height above the moon take to reach the surface of the moon?(You can express the answer as a rational fraction, no need for calculators.)

5.2. DYNAMICS 57

Problem 36.

problems/force-sa-free-fall-cliff.tex

v

t

A ball of mass m is dropped from rest over the edge of a very tall (kilometerhigh) cliff. It experiences a drag force opposite to its velocity of Fd = −bv2.

a) On the axes above, qualitatively plot its downward speed as function oftime.

b) What is its approximate speed when it hits after falling a long time/distance?


Problem 37.

problems/force-sa-kinematic-graphs-1.tex

654321

−1

1

t (seconds)

F (t)x

The graph above represents the force in the positive x direction Fx(t) appliedto a mass m = 1 kg as a function of time in seconds. The mass begins at restat x = 0. The force F is given in Newtons, the position x is given in meters.

a) What is the acceleration of the mass during the time interval from t = 0to t = 6 seconds (sketch a curve)?

b) How fast is the mass going at the end of 6 seconds?

c) How far has the mass travelled at the end of 6 seconds?

5.2. DYNAMICS 59

Problem 38.

problems/force-sa-pushing-two-blocks-2.tex

m MF

The figure shows two blocks of mass M and m that are being pushed along ahorizontal frictionless surface by a force of magnitude F as shown. What is themagnitude of the (contact/normal) force that the block of mass m exerts onthe block of mass M?


Problem 39.

problems/force-sa-pushing-two-blocks.tex

M mF

The figure shows two blocks of mass M and m that are being pushed along ahorizontal frictionless surface by a force of magnitude F as shown. What is themagnitude of the (contact/normal) force that the block of mass M exerts onthe block of mass m?

5.2. DYNAMICS 61

Problem 40.

problems/force-sa-two-blocks-friction.tex

1

m2

m

F

µs µk,

A block of mass m1 is placed on a larger block of mass m2 > m1, where thereis a coefficient of static friction µs = 0.25 and a coefficient of kinetic frictionµs = 0.2 for the surface in contact between the blocks. Both blocks are on africtionless table. A force of magnitude F = 3(m1 + m2)g is applied to thebottom block only.

a) Find the acceleration of the top block only (magnitude and direction).

b) Is the magnitude of the acceleration of the lower block greater than, lessthan, or equal to the accleration of the upper block?


5.2.4 Long Problems

Problem 41.

problems/force-pr-a-constant-simplest.tex

x

y

x 0

v0m

A block of mass m sits on a horizontal frictionless table as shown. A constantforce ~F = F x in the +x-direction (to the right) is applied to it. The mass isinitially moving to the left with speed v0, and starts a the position x0 as shown.

a) Draw a force diagram for the mass m onto the figure above. This shouldinclude all the forces, including those that cancel.

b) Write down an expression for the acceleration ~a of the mass.

c) Integrate the acceleration one time to find ~v(t).

d) Integrate the velocity one time to find ~x(t).

e) How long will it take to bring the particle to rest (where infinity is apossible answer)?

f) Where will it be when it comes to rest?

5.2. DYNAMICS 63

Problem 42.

problems/force-pr-atwoods-machine.tex

m1 m2

In the figure above Atwood’s machine is drawn – two masses m1 and m2 hangingover a massless frictionless pulley, connected by a massless unstretchable string.

a) Draw free body diagrams (isolated diagrams for each object showing justthe forces acting on that object) for the two masses in the figure above.

b) Convert each free body diagram into a statement of Newton’s Second Lawfor that object.

c) Find the acceleration of the system and the tensions in the string on both

sides of the pulley in terms of m1, m2, and g.

d) Suppose mass m2 > m1 and the system is released from rest with themasses at equal heights. When mass m2 has descended a distance H , findthe speed of the masses.


Problem 43.

problems/force-pr-basketball-trajectory.tex

0v

H

Rθ

x

y

A basketball player goes up for a 3-pointer at a (horizontal) distance R fromthe basket. To shoot over his opponent’s outstretched arm, he releases thebasketball of mass m at an angle θ with respect to the horizontal, at a heightH below the height of the rim.

We would like to find v0, the speed he must release the basketball with to sinkhis shot. Assuming that his release is on line and undeflected at initial speedv0, answer the following questions in terms of the givens.

a) Find the vector acceleration (magnitude and direction) of the basketballafter it is released from Newton’s Second Law.

b) Find the vector position of the basketball as a function of time, e.g. x(t)and y(t) in the coordinate system shown.

c) What must v0 be (in terms of H , R, g and θ) for the ball to go throughthe hoop “perfectly” as shown?

5.2. DYNAMICS 65

Problem 44.

problems/force-pr-bead-on-hoop.tex

R

m v

A bead of mass m is threaded on a metal hoop of radius R. There is a coefficientof kinetic friction µk between the bead and the hoop. It is given a push to startit sliding around the hoop with initial speed v0. The hoop is located on thespace station, so you can ignore gravity.

a) Find the normal force exerted by the hoop on the bead as a function ofits speed.

b) Find the dynamical frictional force exerted by the hoop on the bead as afunction of its speed.

c) Find its speed as a function of time. This involves using the frictional forceon the bead in Newton’s second law, finding its tangential acceleration onthe hoop (which is the time rate of change of its speed) and solving theequation of motion.

All answers should be given in terms of m, µk, R, v (where requested) and v0.


Problem 45.

problems/force-pr-bead-on-semicircular-hoop.tex

m

θ R

ω

A small frictionless bead is threaded on a semicircular wire hoop with radius R,which is then spun on its vertical axis as shown above at angular velocity ω.

a) Find the angle θ where the bead will remain stationary relative to therotating wire as a function of R, g, and ω.

b) From your answer to the previous part, it should be apparent that thereis a minimum angular velocity ωmin that the hoop must have before thebead moves up from the bottom at all. What is it? (Hint: Think aboutwhere the previous answer has solutions.)

5.2. DYNAMICS 67

Problem 46.

problems/force-pr-bombing-run-trajectory-easy.tex

0

θt

v

H

A bomber flies with a constant horizontal velocity v0. It wishes to target adummy tank on the ground in a practice bombing run. The bombardier willdrop a bomb (of mass m) when the view angle of the tank relative to the bomberis θt. The bomber’s height is H . Assume that there are no drag forces.

What should the angle θt be at the instant of release if the bomber wishes tohit the tank?


Problem 47.

problems/force-pr-bombing-run-trajectory.tex

0

θt

v

H

A bomber flies with a constant horizontal velocity v0. It wishes to target adummy tank on the ground in a practice bombing run. The bombardier willdrop a bomb (of mass m) when the view angle of the tank relative to the bomberis θt. The bomber’s height is H . Assume that there are no drag forces.

a) What should the angle θt be at the instant of release if the bomber wishesto hit the tank?

b) What is the speed of the the bomb when it hits the tank?

5.2. DYNAMICS 69

Problem 48.

problems/force-pr-dropping-washington-duke-trajectory.tex

D

H

ROSES

M

θh

v0

me

student

A physics student irritated by the personal mannerisms of their physics professordecides to rid the world of him. The student plans to drop a large, massive object(the statue of Washington Duke, actually, recently stolen by pranksters from hisfraternity), mounted on nearly frictionless casters, from a tall building of heightH with a smooth roof sloped at the angle θ as shown. However, the student(being a thoughtful sociopath) wants to make sure that the mass M will make itover the roses to the path a distance D from the base of the building and needsto know how far to let the statue roll down the roof to get the right speed.

Unfortunately, the student isn’t very good at physics and comes to you for help.Since they don’t want to tell you which building or which path they want touse (you might be able to testify against them!) they want you to find (in twosteps, each counting as a separate problem) a general formula for the requisitedistance.

a) Help them out. Start by finding v0 in terms of H , M , D, θ and g (thegravitational constant) that will drop M on RGB assuming no friction or

drag forces. (That way I’m still pretty safe).

b) Now that you know the speed (or rather, assuming that you know thespeed, as the case may be) find h (the vertical distance the statue must


roll down, released from rest, to come off with the right speed). Explicitlyshow that your overall answer (in which v0 should NOT appear) has theright units. If you were clueless in problem 4) you may leave v0 in youranswer but should still try to find SOME combination of the letters H ,M , D, θ and g that has the right units and varies the way you expect theanswer to (more height H means smaller h, for example, so it probablybelongs on the bottom).

5.2. DYNAMICS 71

Problem 49.

problems/force-pr-evie-kniebel-trajectory.tex

θ = 37.5o v0

1m

4m

ramp

Evie Kniebel is a stunt woman for a movie is trying to jump a motorcycle acrossa ditch and land on a special ramp that is built into the branch of a convenienttree. The horizontal gap she must leap to reach the ramp is 4 m. The ramp isvertically 1 m above the lip of the takeoff ramp. The angle of the takeoff rampis fixed at 37.5 (which just happens to be the angle of a 345 right triangle).

The “plot” of the movie requires that the ditch be filled with 16 to 20 footmugger crocodiles imported from the Ganges in India just for this film, as thedirector is a sucker for realism and doesn’t believe in computer generated crocsor latex croc-bots. Feeding them is cheap, however – so far they’ve dined onEvie’s stunt-cousins Evel, Weevel, and Abel Kneibel...

If Evie jumps even a bit too high, she will wreck on an overhanging branch andthe crocs will dine well yet again. If she jumps too low, she bounces back andalso becomes crocodile-bait. With what speed v0 must she take off to completethe jump “just right” (and live to get paid)? Note that you must answer thequestion algebraically FIRST and only then worry about numbers.

And you thought physics wasn’t useful...


Problem 50.

problems/force-pr-flat-plane-three-blocks.tex

M

2M

3M

T1 T2

Three blocks of mass M , 2M and 3M are drawn above. The middle block (2M)sits on a frictionless table. The other two blocks are connected to it by masslessunstretchable strings that run over massless frictionless pulleys. At time t = 0the system is released from rest. Find:

a) The acceleration of the middle block sitting on the table.

b) The tensions T1 and T2 in the strings as indicated.

5.2. DYNAMICS 73

Problem 51.

problems/force-pr-flat-plane-two-blocks-friction.tex

m2

1m

A mass m1 is attached to a second mass m2 by an Acme (massless, unstretch-able) string. m1 sits on a table with which it has coefficients of static anddynamic friction µs and µk respectively. m2 is hanging over the ends of a table,suspended by the taut string from an Acme (frictionless, massless) pulley. Attime t = 0 both masses are released.

1. What is the minimum mass m2,min such that the two masses begin to move?

2. If m2 = 2m2,min, determine how fast the two blocks are moving when massm2 has fallen a height H (assuming that m1 hasn’t yet hit the pulley)?


Problem 52.

problems/force-pr-flat-plane-two-blocks.tex

m2

1m

A mass m1 is attached to a second mass m2 by an Acme (massless, unstretch-able) string. m1 sits on a frictionless table; m2 is hanging over the ends of atable, suspended by the taut string from an Acme (frictionless, massless) pulley.At time t = 0 both masses are released.

a) Draw the force/free body diagram for this problem.

b) Find the acceleration of the two masses.

c) How fast are the two blocks moving when mass m2 has fallen a height H(assuming that m1 hasn’t yet hit the pulley)?

5.2. DYNAMICS 75

Problem 53.

problems/force-pr-golf-on-moon.tex

vf = ?

t = ?

H

v0

R = ?

f

An astronaut on the moon hits a golf ball of mass m horizontally from a teeH meters above the plane as shown. The initial speed of the ball is v0 in thex-direction only. The gravitational force law for the moon is:

~F m = −mg

6j

Note that there are no drag forces as the moon is in a vacuum, and that thelunar plane is flat on the scale of this picture. Use Newton’s second law toanswer the following questions:

a) How long does it take the ball to reach the ground?

b) How far from the base of the cliff where the tee is located does the ballstrike?

c) How fast is the ball going when it hits the ground?


Problem 54.

problems/force-pr-inclined-atwoods-machine-downhill-friction.tex

M

M

2

θ

1

Two blocks with mass M1 and M2 are arranged as shown with M1 sitting on aninclined plane and connected with a massless unstretchable string running overa massless, frictionless pulley to M2, which is hanging over the ground. Thetwo masses are released initially from rest. The inclined plane has coefficientsof static and kinetic friction µs and µk respectively where the angle θ is smallenough that mass M1 would remain at rest due to static friction if there wereno mass M2.

a) Draw separate free-body diagrams for each mass M1 and M2, and se-lect (indicate on your figure) an appropriate coordinate system for eachdiagram;

b) Find the minimum mass M2,min such that the two masses begin to move;

c) If M2 is some value greater than this minimium (so that the block defi-nitely slides), determine the magnitude of the acceleration of the blocks.

5.2. DYNAMICS 77

Problem 55.

problems/force-pr-inclined-plane-three-blocks-2.tex

2MT

T

1

2

θ

3M

M

Three blocks of mass M , 2M and 3M are drawn above. The middle block (2M)sits on a frictionless table tipped at an angle θ with the horizontal as shown.The other two blocks are connected to it by massless unstretchable stringsthat run over massless frictionless pulleys. At time t = 0 the system isreleased from rest. Find:

a) The magnitude of the acceleration of the middle block sitting on the in-clined table.



Problem 56.

problems/force-pr-inclined-plane-three-blocks.tex

M

2M

3M

T

T

1

2

θ

Three blocks of mass M , 2M and 3M are drawn above. The middle block (2M)sits on a frictionless table tipped at an angle θ with the horizontal as shown.The other two blocks are connected to it by massless unstretchable strings thatrun over massless frictionless pulleys. At time t = 0 the system is released fromrest. Find:

a) The acceleration vector in Cartesian components of the middle blocksitting on the table. Any correct way of uniquely specifying the Cartesianvector will be accepted.


5.2. DYNAMICS 79

Problem 57.

problems/force-pr-inclined-plane-two-blocks-1.tex

θ

2M

M

Two blocks of mass M and 2M are drawn above. The block 2M sits on a fric-tionless table tipped at an angle θ with the horizontal as shown. The other blockis connected to it by massless unstretchable string that runs over a massless,frictionless pulley. At time t = 0 the system is released from rest. Find:

a) The acceleration (magnitude and direction as in up or down the incline)of the block on the incline.

b) The tension T in the string.

c) At what angle θ do the blocks not move?


Problem 58.

problems/force-pr-inclined-plane-two-blocks.tex

mm

θ

Two blocks of mass m sit on a frictionless surface and are connected by amassless, unstretchable string that rolls on a massless, frictionless pulley asshown. The second mass sits on an inclined tipped at an angle θ relative tohorizontal. At time t = 0, both masses are released from rest.

a) Find the acceleration of both masses (magnitude, but indicate directionsfor each mass on the figure).

b) Find the tension T in the string.

All answers should be in terms of m, g, and θ.

5.2. DYNAMICS 81

Problem 59.

problems/force-pr-kinematic-graphs.tex

-20

-15

-10

-5

0

5

10

15

20

0 2 4 6 8 10

x (m)

t (seconds)

-10

-5

0

5

10

0 2 4 6 8 10

v (m/sec)

t (seconds)

-10

-5

0

5

10

0 2 4 6 8 10a (m/sec2)

t (seconds)

On the figure above a trajectory x(t) for one-dimensional motion is plotted inthe top panel. In the second and third panel sketch in approximate curves thatrepresent v(t) and a(t) respectively. Your curves should at least qualitatively

agree with the top figure and correctly identify ranges of positive, negative andzero velocity and acceleration.


Problem 60.

problems/force-pr-monkey-gun.tex

θ0v

D

H

A hunter aims his gun directly at a monkey in a distant tree. Just as she fires,the monkey lets go and drops in free fall towards the ground. Show that thebullet hits the monkey.

5.2. DYNAMICS 83

Problem 61.

problems/force-pr-pulling-rough-blocks-on-rough-table.tex

µ k

µs

M

m

θ

F

A rope at an angle θ with the horizontal is pulled with a force vF . It pulls,in turn, two blocks, the bottom with mass M and the top with mass m. Thecoefficients of friction are µs between the top and bottom block (assume that

they do not slide for the given force ~F ) and µk between the bottom blockand the table. Remember to show (and possibly evaluate) all forces acting onboth blocks, including internal forces between the blocks.

a) Draw a “free body diagram” for each mass shown, that is, draw in andlabel all real forces acting on it;

b) Apply Newton’s Second Law in appropriate coordinates to each massshown;

c) Solve for the acceleration(s) of each mass shown and evaluate all unknownforces (such as a normal force or the tension in a string) in terms of thegiven quantities.

Don’t forget that the acceleration is a vector and must be given as a magnitudeand a direction (for example, “along the plane to the right” is ok) or in vectorcomponents.


Problem 62.

problems/force-pr-pushing-vertical-blocks-friction.tex

MmF

A force ~F is applied to a large block with mass M , which pushes a smaller blockof mass m as shown. The large block is supported by a frictionless table. Thecoefficient of static friction between the large block and the small block is µs.Find the magnitude of the minimum force Fmin such that the small block doesnot slide down the face of the large one. Draw free body diagrams and show allof your reasoning.

5.2. DYNAMICS 85

Problem 63.

problems/force-pr-range-of-cannon-on-hill.tex

m

ymax

θ

ov

H

R

A cannon sits on at the top of a rampart of height (to the mouth of the cannon)H . It fires a cannonball of mass m at speed v0 at an angle θ relative to theground. Find:

a) The maximum height ymax of the cannonball’s trajectory.

b) The time the cannonball is in the air.

c) The range of the cannonball.


Problem 64.

problems/force-pr-terminal-velocity-tom-and-jerry.tex

The script calls for Tom (cat) to chase Jerry (mouse) across the top of a cartoonskyscraper of height H and off the edge where they both fall straight down (theirinitial x-velocity is negligible as they fall off) towards a soft pile of dirt that willkeep either one from getting hurt by the fall no matter how hard they land (notoon animals were injured in this problem).

Your job is to work out the physics of a “realistic” fall for the animation team.You decide to use the following for the drag force acting on either one:

~Fi = −biv2v

where hv is a unit vector in the direction of the velocity and i = t, j for Tom orJerry respectively and where:

bi = CL2i

mi = DL3i

(that is, the drag force is proportional to their cross-sectional area and theirmass is proportional to their volume). Their relative size is Lt = 5Lj (Tom isfive times the height of Jerry).

a) Draw a on the back of the preceding page showing the building, Tom andJerry at the instant that Tom runs off of the top. Jerry (who is ahead)should have fallen a short distance d towards the ground.

b) Using the laws of physics, determine the equation of motion (find an ex-pression for the acceleration and write it as a differential equation) al-gebraically (so the solution applies to Tom or Jerry equally well). Youranswer can be given in terms of bi and mi.

c) Without solving the equation of motion, find an algebraic expression forthe terminal velocity of Tom or Jerry as functions of Li. Explain/showyour reasoning, don’t just write down an answer.

5.2. DYNAMICS 87

Problem 65.

problems/force-pr-triple-atwoods-machine.tex

H

M M

m

A block and tackle arrangement is built with three massless pulleys and threehanging masses with masses M , m, and M as shown above. The two M massesare a height H off the ground, and m is on the ground. At time t = 0 the massesare released from rest from this configuration.

a) Draw a GOOD free body diagram. Clearly label all quantities.

b) Find the acceleration (magnitude and direction) of each block and thetension T in the string as a function of the givens, assuming that M +M >m.

c) Find the velocity of each block when the blocks of mass M hit the ground.


Problem 66.

problems/force-pr-two-balls-1D-2.tex

H

v02

m

mt = t 0

v1

A ball of mass m is dropped at time t = 0 from rest (v01 = 0) from the topof the Duke Chapel (which has height H) to fall freely under the influence ofgravity. A short time t = t0 later a second ball, also of mass m, is thrown downafter it at speed v02. Neglect drag.

a) (2 points) Draw a free body diagram for and compute the net force actingon each mass separately.

b) (4 points) From the equation of motion for each mass, determinetheir one dimensional trajectory functions, y1(t) and y2(t).

c) (3 points) Sketch qualitatively correct graphs of y1(t) and y2(t) on thesame axes in the case where the two collide.

5.2. DYNAMICS 89

Problem 67.

problems/force-pr-two-blocks-on-inclined-plane-plus-pulley.tex

3Mθ

T

2M

M

T1

2

Three blocks of mass M , 2M and 3M are drawn above. The two smaller blockssit on a frictionless table tipped at an angle θ with respect to the horizontal asshown. The three blocks are connected by massless unstretchable strings, oneof which runs over a massless frictionless pulley to the largest mass. At timet = 0 the system is released from rest. Find:

a) The acceleration vector in Cartesian components of the middle block onthe incline. Any correct way of uniquely specifying the Cartesian vectorwill be accepted, for example ~a = (ax, ay), or ~a = axx + ayy.

b) The magnitude of the tensions T1 and T2 in the strings as indicated onthe diagram.


Problem 68.

problems/force-pr-two-blocks-with-friction.tex

kµ

m

MF

,µs

frictionless

A small block of mass m sits on top of a large block of mass M that sits ona frictionless table. The coefficient of static friction between the two blocks isµs and the coefficient of kinetic friction between the two blocks is µk. A force~F = F x is directly applied to the lower block as shown. All answers should begiven in terms of m, M , µs, µk, and g.

a) What is the largest force Fmax that can be applied such that the upperblock does not slide on the lower block?

b) Suppose that F = 2Fmax (so that the upper block slips freely). What isthe acceleration of each block?

5.2. DYNAMICS 91

Problem 69.

problems/force-pr-two-masses-incline-different-friction.tex

m

m

1

2

θ

Two blocks, each with the same mass m but made of different materials, siton a rough plane inclined at an angle θ such that they will slide (so that thecomponent of their weight down the incline exceeds the maximum force exertedby static friction). The first (upper) block has a coefficient of kinetic friction ofµk1 between block and inclined plane; the second (lower) block has coefficient ofkinetic friction µk2. The two blocks are connected by a massless unstretchablestring.

Find the acceleration of the two blocks a1 and a2 down the incline:

a) when µk1 > µk2;

b) when µk2 > µk1.


Problem 70.

problems/force-pr-vertical-block-friction.tex

Fm

µs

M

A block of mass M is pushed on a frictionless table by a force ~F to the right. Amass m is positioned on the front face as shown. There is a coefficient of staticfriction µs between the big and little block. Find the minimum magnitude offorce Fmin that will keep the little block from slipping down the big one.

5.3. CIRCULAR MOTION 93

5.3 Circular Motion


Problem 71.

problems/circular-motion-mc-ball-on-string-breaks.tex

direction of rotationE D

CA

B

string breaks here

A ball is being whirled on a string. At the instant shown, the string breaks.Select the correct trajectory of the ball after it breaks.

a) A

b) B

c) C

d) D

e) E


Problem 72.

problems/circular-motion-mc-two-masses-on-strings-qual.tex

m

ω

LL

m TT1 2

A block of mass m is tied to a cord of length L that is pivoted at the center ofa frictionless table. A second block of mass m is tied to the first block also ona cord of length L, and both are set in motion so that they rotate together atangular speed ω as shown above. The tensions T1 and T2 in the cords are:

a) T1 = T2

b) T1 > T2

c) T1 < T2

d) T1 > T2 for ω > 0 and T1 < T2 for ω < 0


Problem 73.

problems/circular-motion-mc-two-masses-on-strings.tex

m

ω

LL

m TT1 2

A block of mass m is tied to a cord of length L that is pivoted at the center ofa frictionless table. A second block of mass m is tied to the first block also ona cord of length L, and both are set in motion so that they rotate together atangular speed ω as shown above. The tensions T1 and T2 in the cords are:

a) T1 = 3mω2L, T2 = 2mω2L

b) T1 = mω2L, T2 = 2mω2L

c) T1 = mω2L, T2 = mω2L

d) T1 = 2mω2L, T2 = 2mω2L

e) T1 = mω2L, T2 = 4mω2L


5.3.2 Long Problems

Problem 74.

problems/circular-motion-pr-conic-pendulum-tether-ball.tex

v

θ

A tether ball of mass m is suspended by a rope of length L from the top ofa pole. A youngster gives it a whack so that it moves in a circle of radiusr = L sin(θ) < L around the pole. Find an expression for the speed v of theball as a function of θ.


Problem 75.

problems/circular-motion-pr-puck-on-cylinder-friction.tex

rr

mg (in)

Side View Top View

g (down)

(out)

m

z

Ω

Ω

A hockey puck with mass m is placed against the wall of a vertical rotatingcylinder of radius r shown in side and overhead views above. The coefficientof static fraction between the puck and the wall of the cylinder is µs. Gravitypoints down in the left hand figure and into the page in the right hand figurein this problem as shown. The cylinder is rotating at a constant angularvelocity.

a) On the diagrams above draw in all forces that act on the mass whilethe cylinder rotates at constant angular speed. For forces acting vertically,use the side view. For forces acting in the horizontal directions, use thetop view.

b) Write down Newton’s second law for both the vertical and radial di-rections.

c) What is the minimum angular speed Ωmin such that the hockey puck doesnot slide down the wall?


Problem 76.

problems/circular-motion-pr-puck-on-wheel-friction.tex

M

m

ω

µ s

r

Diskhockey puck

block

A disk is rotating with a constant angular velocity ω. A small hockey puck ofmass m is placed on the disk at a distance r from the center, and is attached toanother block with mass M hanging below by a massless unstretchable stringthat passes through a tiny (frictionless) hole right in the center of the disk. Thestatic friction coefficient between the hockey puck m and the disk is µs, andµsmg < Mg. Find the following:

a) The smallest M that will not move down (as the puck slips on the disk).

b) The largest M that will not move up (as the puck slips on the disk).


Problem 77.

problems/circular-motion-pr-rounding-a-banked-curve-frictionless.tex

R

v towards center

θ

(top view) (side view)

m

A car of mass m is rounding an icy frictionless banked curve that has radiusof curvature R and banking angle θ. What must the speed v of the car be suchthat it can succeed in making it around the curve without sliding off of the roaduphill or down?


Problem 78.

problems/circular-motion-pr-rounding-a-banked-curve.tex

R

v towards center

θ


m

A car of mass m is rounding a banked curve that has radius of curvature R andbanking angle θ. The coefficient of static friction between the car’s tires andthe road is µs. Find the range of speeds v of the car such that it can succeed inmaking it around the curve without skidding.


Problem 79.

problems/circular-motion-pr-rounding-a-flat-curve.tex

R

v


towards centerm

(9 points) A car of mass m is rounding a flat (unbanked) curve that has radiusof curvature R. The coefficient of static friction between the car’s tires and theroad is µs. Find the fastest speed of the car v such that it can succeed in makingit around the curve without skidding.

Chapter 6

Work and Energy

In the previous problems, you were frequently asked to e.g. find the speed atthe bottom of an incline or after a mass falls from some height. Every singletime, you had to find the time it reached the bottom and substitute it back intothe expression for the velocity/speed.

Work (and the general concept of energy) arise when we eliminate time onceand for all from Newton’s Laws, and directly relate speed to position. Atfirst this is just a convenience that simplifies certain kinds of problems. Later,however, we will see that energy is in some sense more fundamental thanforce, that we could have started our study of dynamics from the beginningwith energy concepts and worked from various potential energies to forces.

Sadly, certain non-conservative forces are difficult to connect back to potentialenergies in laws of nature – they do come from them but we’ll only understandthat after the fact. In the meantime, it makes more sense to go from force toenergy than from energy to force, although we’ll do a bit of both in the followingproblems.

When you you think maybe I should use work or energy...” in a problem?

Whenever the questions asked relate speed to position independent of time (or,of course, when the questions asked directly pertain to the concept of work andenergy itself). How fast at the bottom of the incline? Think energy. How highdoes a mass thrown upward rise before stopping? think energy.

103

104 CHAPTER 6. WORK AND ENERGY

6.1 Work and Kinetic Energy


Problem 80.

problems/wke-mc-block-and-friction-2.tex

µk

F

m θ

A block of mass m is on a floor. The kinetic friction coefficient between the blockand the floor is µk. A student pulls a block with a force ~F directed upward atan angle θ with respect to the horizontal as shown. What is the work done byfriction when the block moves a distance L along the floor to the right?

a) −µkmgL

b) −µk(mg − F sin(θ))L

c) FL cos(θ) − µkmgL + µkF sin(θ)L

d) FL cos(θ)

e) −µk(mg + F sin(θ))L

6.1. WORK AND KINETIC ENERGY 105

Problem 81.

problems/wke-mc-block-and-friction.tex

F

µθ km

A block of mass m is on a floor. The kinetic friction coefficient between theblock and the floor is µk. A student pushes a block with a force ~F directeddown at an angle θ with respect to the horizontal as shown. What is the workdone by the student (the force ~F ) when the block moves a distance L alongthe floor?

a) FL

b) µkmgL

c) FL cos(θ)

d) µk(mg + F sin(θ))L



Problem 82.

problems/wke-mc-cannonball-energy-speed-trajectories.tex

a b

v0

0v

Two cannons fire cannonballs at the same initial speed v0 into the air along thetrajectories shown. Neglect the drag force of the air. Which cannonball strikesthe ground faster?

a) Cannonball a hits going faster.

b) Cannonball b hits going faster.

c) Cannonball a and b hit at the same speed

d) We cannot tell which hits the ground going faster without more informa-tion than is given in the problem and picture.



Problem 83.

problems/wke-ra-work-by-force-2.tex

d

FA) M

d

B) MF

θ

In the figure above a force with a constant magnitude F is applied to a blockof mass M resting on a table with a rough surface at two different angles asshown. The coefficient of kinetic friction between the block and table is µk. Asthe block slides to the right a distance d, the work done by ~F is WF , and thework done by friction is Wfk

.

a) Rank the magnitude of the work done by ~F in two cases, WAF and WB

F ;

b) Rank the magnitude of the work done by friction in two cases, WAfk

and

WBfk

.


Problem 84.

problems/wke-ra-work-by-force-3.tex

d

FA) M

d

B) M

θF

In the figure above a force with a constant magnitude F < Mg is applied to ablock of mass M resting on a table with a rough surface at two different anglesas shown. The coefficient of kinetic friction between the block and table is µk.As the block slides to the right a distance d, the work done by ~F is WF , andthe work done by friction is Wfk

.

a) Rank the magnitude of the work done by ~F in two cases, WAF and WB

F ;

b) Rank the magnitude of the work done by friction in two cases, WAfk

and

WBfk

.


Problem 85.

problems/wke-ra-work-by-force.tex

A

B

C

M θθ

d

In the figure above a force with a constant magnitude F is applied to a blockof mass M resting on a smooth (frictionless) table at three different angles as

shown. Rank the work done by ~F as the block slides to the right a distanced, where equality is allowed. (A possible answer might be A = B > C forexample.)


6.1.3 Short Answer

Problem 86.

problems/wke-sa-block-on-paper.tex

mµk

v

D

A block of mass m is initially at rest on a long piece of smooth paper on africtionless table. The block has a coefficient of kinetic friction µk with thepaper. You pull the paper horizontally out from under the block quickly inthe direction indicated by the arrow, such that the block moves a distance D(relative to the ground) while still on the paper.

a) What is the magnitude of the work done by kinetic friction on the block?

b) Is the work done positive (increasing the kinetic energy of the block) ornegative (decreasing the kinetic energy of the block).

c) What is the final velocity of the block when it comes off of the paper andslides along the frictionless table? Use +x to the right!


Problem 87.

problems/wke-sa-graph-work-1.tex

F(x)

1

2

3

−1

−2

−3

2 4 6 8 x1 3 5 7 9 1110 120

The graph above represents a force in the positive x direction F (x) applied toa mass m as a function of its position. The mass begins at rest at x = 0. Theforce F is given in Newtons, the position x is given in meters.

a) How much work is done going from x = 0 to x = 6?

b) How much work is done going from x = 6 to x = 12?

c) Assuming m = 1 kg, what is the final velocity of the object at x = 12?


Problem 88.

problems/wke-sa-graph-work-2.tex

x654321

−1

1

F(x)

The graph above represents the total one-dimensional force in the x directionF (x) being applied to a mass m as a function of its position. The mass begins atrest at x = 0 and moves only along the x axis. The force F is given in Newtons,the position x is given in meters.

a) How much work is done going from x = 0 to x = 3?

b) How much work is done going from x = 3 to x = 6?

c) Assuming that m = 1 kg and that it begins at rest at the beginning ofthe motion, does the mass reach x = 6?


Problem 89.

problems/wke-sa-work-done-by-force.tex

θ

d d

F

AF

BM M

A B

In the figure above a force with a constant magnitude FA = FB = F is appliedto a block of mass M resting on a table with a rough surface at two differentangles as shown. The coefficient of kinetic friction between the block and tableis µk. As the block slides to the right a distance d, the work done by ~F is WF ,and the work done by friction is Wfk

in the two cases, A and B.

a) For case A, find the work done by ~F and friction, WAF and WA

fk, respec-

tively. Your answers should have the correct sign.

b) For case B, find the work done by ~F and friction, WBF and WB

fk, respec-

tively. Your answers should have the correct sign.


6.1.4 Long Problems

Problem 90.

problems/wke-pr-painball-gun-exponential-example.tex

D

m

xoF = F e

−x/D

A simple schematic for a paintball gun with a barrel of length D is shown above;when the trigger is pulled carbon dioxide gas under pressure is released into theapproximately frictionless barrel behind the paintball (which has mass m). Theexpanding, cooling gas exerts a force on the ball of magnitude:

F = F0e− x

D

on the ball to the right, where x is measured from the paintball’s initial positionas shown.

a) Find the work done on the paintball by the force as the paintball is accel-erated down the barrel.

b) Use the work-kinetic-energy theorem to compute the kinetic energy of thepaintball after it has been accelerated.

c) Find the speed with which the paintball emerges from the barrel after thetrigger is pulled.


Problem 91.

problems/wke-pr-painball-gun-exponential-solution.tex

D

m

xoF = F e

−x/D

A simple schematic for a paintball gun with a barrel of length D is shown above;when the trigger is pulled carbon dioxide gas under pressure is released into theapproximately frictionless barrel behind the paintball (which has mass m). Theexpanding, cooling gas exerts a force on the ball of magnitude:

F = F0e− x

D





The only force acting on the paintball is the force applied by the prssurizedgas (gravity is countered by a normal force from the barrel, and in anycase neither does work when the motion is horizontal; we are neglectingfriction which may be less realistic). So WKE reads simply

Kf − Ki =

∫ f

i

bfF · dx

=

∫ D

0

F0e− x

D dx

= F0

(

−De−d/D) ∣

∣

∣

D

0

= −FD(e−1 − 1) = FD(1 − 1/e) .

(6.1)

This is the work done by the gas. Since the paintball starts at rest soKi = 0 it is also the kinetic energy the ball has when it leaves the barrel.


To find the speed with which it leaves we set

mv2

2= Kf = FD(1 − 1/e)

v =

√

2FD(1 − 1/e)

m. (6.2)


Problem 92.

problems/wke-pr-paintball-gun-adiabatic.tex

D

m

x

F = (D − x)Fo

D

A simple schematic for a paintball gun is shown above; when the trigger ispulled carbon dioxide gas under pressure is released into the approximatelyfrictionless barrel behind the paintball (which has mass m) initially resting atx0. The gas expands approximately adiabatically and exerts a force on theball of magnitude:

F = F0xγ

0

xγ

on the ball to the right, where F0 is the initial force exerted at x = x0, and xis measured from the end of the barrel as shown. γ is a constant (equal to 1.4for carbon dioxide). This force is only exerted up to the end of the barrel atx = 4x0.





Problem 93.

problems/wke-pr-paintball-gun-linear.tex

D

m

x

F = (D − x)Fo

D

A simple schematic for a paintball gun is shown above; when the trigger is pulledcarbon dioxide gas under pressure is released into the approximately frictionlessbarrel behind the paintball (which has mass m). The gas exerts a force on theball of magnitude:

F =F0

D(D − x)





6.2. WORK AND MECHANICAL ENERGY 119

6.2 Work and Mechanical Energy


Problem 94.

problems/wme-mc-battleship.tex

A battleship simultaneously fires two shells at enemy ships along the trajec-tories shown, such that the shells have the same initial speed. One ship (A)is close by; the other ship (B) is far away.

Which shell has the greater speed when it hits the ship (circle both if the speedsare equal)? Ignore drag or coriolis forces.

A B


Problem 95.

problems/wme-mc-zero-of-potential-energy.tex

Tommy is working on a physics problem involving energy. “Look,” he says, “thetotal energy of this block at rest is zero at the top of this incline of height Hand therefore must be zero at the bottom.”

Sally disagrees. “Impossible. The block is at the top of the incline. It hastotal energy mgH at the top and so its total energy must still be mgH at thebottom.”

a) Tommy is right, Sally is wrong.

b) Sally is right, Tommy is wrong.

c) Both Tommy and Sally are right.

d) Both Tommy and Sally are wrong.

e) There isn’t enough information to tell who is right and who is wrong.



Problem 96.

problems/wme-ra-two-skiers.tex

A B

A

B

start

finish

Two skiers start at the same point on a (frictionless) slope. One (A) skis straightdown the slope to the finish line. The other (B) passively skis off of a ski jumpto arrive at the finish more flamboyantly. Rank the answers to the followingquestions, where equality is a possibility, that is, possible answers are A < B orA = B. Ignore friction and drag forces and assume that the jumper does notuse their leg muscles to “jump”.

a) Rank the relative speed of the two skiers when they reach the finish line.

b) Rank the finish time – who arrives at the finish line first (or is it at thesame time)?


6.2.3 Short Answer

Problem 97.

problems/wme-sa-ball-to-ground.tex

A ball is thrown with some speed v0 from the top of a cliff of height H . Showthat the speed with which it hits the ground is independent of the direction itis thrown (and determine that speed in terms of g, H , and v0).


Problem 98.

problems/wme-sa-potential-energy-to-force.tex

0

−10

−15

−20

U(x

) (J

)0 5 10 15

−5

x (m)

A one-dimensional force F (x) acts on a 2 kg particle which moves along thex axis. The potential energy U(x) associated with F (x) is shown below. Theparticle is at x = 11 m, its speed is 2 m/s.

a) What is the magnitude and direction of F (x) at:x = 11 m:x = −3 m:x = 6 m:

b) Between what limits of x does the particle move?

c) What is its speed at x = 7 m?


Problem 99.

problems/wme-sa-sliding-block-friction.tex

A block of mass m sitting on a horizontal surface is given an initial speed v0.Travelling in a straight line it comes to rest after sliding a distance d. Show

that the coefficient of kinetic friction is given by v2

2gd .


Problem 100.

problems/wme-sa-spring-jumper-straight-up-solution.tex

d

H

m

y

k(d−y)

mg

mg

A simple child’s toy is a jumping frog made up of an approximately masslessspring of uncompressed length d and spring constant k that propels a molded“frog” of mass m. The frog is pressed down onto a table (compressing the springby d) and at t = 0 the spring is released so that the frog leaps high into the air.

Use work and/or mechanical energy to determine how high the frog leaps.

In this problem, all the forces that act (spring, gravity) are conservative somechanical energy is conserved. We will use this to solve the problem, but let’sfirst recall how we get from the WKE theorem to conservation of mechanicalenergy. The theorem states that the total work done on an object over sometime interval is equal to the change in the object’s kinetic energy:

Wi→f = Kf − Ki . (6.3)

The work, in turn, is given by

Wi→f =

∫ f

i

F · dr , (6.4)

where F is the total force acting on the object. When the total force can bedecomposed as F = F1 +F2 + . . . into various forces (for example, each of theseterms might represent the force applied on the object in question by a particularother object) then the total work is the sum of the work done by each of theseindividual forces.

Conservative forces are those for which the total work done on an object movingfrom ri to rf is independent of the route taken to get there and of the speedwith which the route is traversed. In other words, the work is completely de-termined by the initial and final positions. That this is not true for all forces isimmediately clear, think of the work done by friction when an object is movedin a big loop, starting and ending in the same position. When it is true for the


term Fa, we can define a function of position Ua(r), called potential energy oftype a, such that

Wa = −Ua(rf ) + Ua(ri) . (6.5)

Clearly, we are free to add a constant to Ua. Sine only differences of potentialenergy are meaningful, this will have no physical significance. As shown in class,we can define Ua(r) simply as minus the work done in getting to r starting fromsome arbitrary position r0. Since this work does not depend on how we getthere this definition makes sense. Which position we choose to start with isirrelevant as a different choice shifts Ua precisely by an irrelevant constant.Clearly Ua(r0) = 0 so our choice is essentially a choice of a position where thepotential energy vanishes.

If Fa is conservative, we can use (6.5) to move Wa to the right-hand side of(6.3). Suppose, for example, that F1 is conservative. Then we can write

W1 + W2 + W3 + · · · = Kf − Ki

−U1(rf ) + U1(ri) + W2 + W3 + · · · = Kf − Ki

W2 + W3 + · · · = Kf + U1(rf ) − Ki − U1(ri) . (6.6)

We have dropped the work done by F1 from the left-hand side, adding thepotential energy associated to this force to the right-hand side.

If all the forces involved are conservative, we can do this for all of them. Thisleaves zero on the left-hand side, while on the right-hand side we have the change(final minus initial value) of the sum of kinetic energy and all types of potentialenergy present. This sum we call the total mechanical energy and the fact thatit does not change is the conservation of mechanical energy in the presence ofpurely conservative forces.

When non-conservative forces are present, we can still move the work doneby all conservative forces to the right-hand side, defining a total mechanicalenergy. Since the left-hand side is not zero (it includes the work done by allnon-conservative forces) this will not be conserved.

After this long-winded introduction, let’s go about solving the problem. Herewe have two conservative forces, so our total mechanical energy will have threeterms: kinetic energy, gravitational potential energy, and elastic potential en-ergy stored in the spring. I will use the coordinates indicated in the figure;motion is one-dimensional along the y axis.

We need to find the expressions for potential energy. For gravity we have Fg =−mgy so

Wg =

∫ f

i

Fg · dr = −mg

∫ f

i

dy = −mgyf + mgyi . (6.7)

Comparing this to (6.5) we see that

Ug = mgy . (6.8)


Of course, we are free to add a constant to Ug as we please. This is equivalentin this case to our freedom to choose an origin (the height at which we set y = 0and where Ug = 0).

For a spring we have in general Fs = −kx where x is the deviation of the springfrom its uncompressed (and unstretched) length. If we consider a spring lyingalong the x axis and pick x = 0 as the position of the mass attached to its endwhen the spring is uncompressed, then we can write Fs = −kxx and the workdone by the spring in moving the mass from xi to xf is

Ws =

∫ f

i

Fs · dr =

∫ xf

xi

−kxdx = −kx2

f

2+ k

x2i

2. (6.9)

Comparing again to (6.5) we see that

Us = kx2

2. (6.10)

As aways we can shift this by a constant. This form is somewhat natural as itgives zero potential energy when the spring is uncompressed and unstretched.

Now we can put this all together to write the total energy of our frog. To getthis right we need to pay attention to our coordinates. y designates the heightof the frog above the ground. When y ≤ d this means the spring is compressedfrom its uncompressed length d to length y (i.e. the deviation is x = d − y).Once the height is y > d, the spring, which is not attached to the ground butto the frog, does not become stretched but rather remains uncompressed. Thusthe expression for total mechanical energy is

E =

mv2

2 + mgy + k(d−y)2

2 y ≤ dmv2

2 + mgy y > d .(6.11)

Throughout the frog’s motion, as y and v change, this expression has a constantvalue. This is the conservation of energy. To find the frog’s maximal height, wecompute the total energy at t = 0. Here the frog is at rest, jsut prior to beginningits jump, at height yi = 0. Kinetic and gravitational potential energy vanish,

but the spring is compressed to length zero and we have Ei = kd2

2 . Comparethis to the total energy when the frog has reached the top of its jump at yf = h.At the top of the jump we again have v = 0 so kinetic energy vanishes. Thespring is now uncompressed, so the total energy is simply Ef = mgh. Thesetwo expressions must have the same value, so we find

mgh =kd2

2

h =kd2

2mg. (6.12)


Problem 101.

problems/wme-sa-spring-jumper-straight-up.tex

H

m

A simple child’s toy is a jumping frog made up of an approximately masslessspring with spring constant k that propels a molded “frog” of mass m. The frogis pressed down onto a table (compressing the spring by a distance d) and att = 0 the spring is released so that the frog leaps high into the air.

Use work and/or mechanical energy to determine how high the frog leaps. Ne-glect drag forces.


Problem 102.

problems/wme-sa-tension-pendulum-height-R.tex

θ = π/2R

m

In the figure above, a mass m is attached to a massless unstretchable stringof length R and held at an initial position at an angle θ = π/2 relative to thehorizontal as shown. At time t = 0 it is released from rest. Find the tension Tin the string when it reaches θ = 0.


6.2.4 Long Problems

Problem 103.

problems/wme-pr-loop-the-loop-block.tex

H

R

θ

A block of mass M sits at the top of a frictionless loop-the-loop of height H .

a) Find the normal force exerted by the track when the mass is at an angleθ on the loop as shown.

b) Find the minimum height H such that the block loops the loop withoutcoming off of the track.


Problem 104.

problems/wme-pr-loop-the-loop-classic-example.tex

H

R

θ

A block of mass M sits at the top of a frictionless hill of height H leading to acircular loop-the-loop of radius R.

a) Find the minimum height Hmin for which the block barely goes aroundthe loop staying on the track at the top. (Hint: What is the condition onthe normal force when it “barely” stays in contact with the track? Thiscondition can be thought of as “free fall” and will help us understandcircular orbits later, so don’t forget it.).

Discuss within your recitation group why your answer is a scalar numbertimes R and how this kind of result is usually a good sign that your answeris probably right.

b) If the block is started at this position, what is the normal force exertedby the track at the bottom of the loop, where it is greatest?

If you have ever ridden roller coasters with loops, use the fact that yourapparent weight is the normal force exerted on you by your seat if you

are looping the loop in a roller coaster and discuss with your recitationgroup whether or not the results you derive here are in accord with yourexperiences. If you haven’t, consider riding one aware of the forces thatare acting on you and how they affect your perception of weight andchange your direction on your next visit to e.g. Busch Gardens to be, in abizarre kind of way, a physics assignment. (Now c’mon, how many classeshave you ever taken that assign riding roller coasters, even as an optionalactivity?:-)


Problem 105.

problems/wme-pr-loop-the-loop-classic-solution.tex

a

a

N

mg

mg N

vb

tv

H

R


a) Find the minimum height Hmin for which the block barely goes aroundthe loop staying on the track at the top. (Hint: What is the condition onthe normal force when it “barely” stays in contact with the track? Thiscondition can be thought of as “free fall” and will help us understandcircular orbits later, so don’t forget it.).

Discuss within your recitation group why your answer is a scalar numbertimes R and how this kind of result is usually a good sign that your answeris probably right.


If you have ever ridden roller coasters with loops, use the fact that yourapparent weight is the normal force exerted on you by your seat if you

are looping the loop in a roller coaster and discuss with your recitationgroup whether or not the results you derive here are in accord with yourexperiences. If you haven’t, consider riding one aware of the forces thatare acting on you and how they affect your perception of weight andchange your direction on your next visit to e.g. Busch Gardens to be, in abizarre kind of way, a physics assignment. (Now c’mon, how many classeshave you ever taken that assign riding roller coasters, even as an optionalactivity?:-)

Let us follow the hint and think about what is going on here. In this problemthe block is not bound to the looping track. This means that when it goes overthe top of the loop nothing is “holding it up.” Like any other object not heldup by anything, it must accelerate down with an acceleration g. Yet experience


with toy cars, roller coasters, and strings tells us that if it is going fast enoughit will not fall off the track. The reason is that going around a circular trackdoes involve an acceleration, towards the center of the circle, of magnitude v2/Rwhere v is the speed. We will reproduce this in the last problem on this set. Ifthis acceleration is at least g then at the top of the track the block can be in freefall without leaving the track. If the speed is higher, the acceleration requiredto complete the circle will be higher than g. This means if the track broke, theblock would in fact fly off above the circular trajectory. This is prevented bya normal force applied by the track. The point of all this is that if the blockis moving too slowly around the loop it will leave the track. As the speed isreduced past this minimum, the first failure to stay on the track will occur atthe very top of the loop. This is intuitively clear, we will work it out in detailin another problem.

To turn these words into equations, consider applying Newton’s second law tothe block at the instant when it is at the apex of the looping track, moving(horizontally, to the left) at a speed vt. The figure indicates forces, velocity, andacceleration at this point, including the initially unknown normal force appliedby the track. Newton’s second law then reads

F = −mgy − N y = ma . (6.13)

In order for the block to continue its circular motion along the track this down-ward vertical acceleration must be equal to the centripetal acceleration, directeddownward towards the center of the circle, i.e. we have a = −mv2

t /R y. Thisrequires

N =mv2

t

R− mg . (6.14)

Since N ≥ 0 we see that if the block is moving too slowly it will not stay on thetrack. The minimum speed needed to just maintain contact with the track atthe top is the speed at which N = 0, i.e.

v2t = gR . (6.15)

Now our job is to find how high the initial ramp must be in order for the block toreach the top of the look with this speed. Of course, as it goes down the ramp theblock accelerates under the influence of gravity, but as it goes up the loopingtrack it slows down under the same influence. Since all forces acting on theblock are conservative (gravity) or do no work at all (the normal forces, whichare everywhere perpendicular to the direction of motion) the total mechanicalenergy of the block is conserved throughout its travels along our track. Wecan thus relate its speed at the top of the loop to the height of the rampwhere it was released from rest by equating the total mechanical energy in bothconfigurations, including kinetic and gravitational potential energy. SettingUg = 0 at the base of the loop to determine the irrelevant additive constant wehave for these initial and final configurations the following expressions for total


energy

Ei = mgH

Et =mv2

t

2+ mg 2R . (6.16)

Setting these equal to each other we find that the speed of the block at the topof the loop is determined by H, R as

v2t = g(2H − 4R) . (6.17)

The minimum H needed to clear the loop will lead to a value for v equal to theminimal value (6.15) i.e.

g(2H − 4R) = gR

H =5

2R . (6.18)

As predicted, we find a number, determined by various geometric factors, timesR. This makes sense, because R is the only parameter in the problem with theright dimensions, length. So to determine a length H related in some way to Rwe expect to find a result like this. That it makes sense does not make it trivial.Neglecting friction, we predict that to double the height of the loop you mustdouble the height of your ramp. And we could have predicted that just usingthis kind of dimensional reasoning, without doing any calculations at all!

We now want to find the normal force applied by the track at the bottom of theloop when the block is released from this height. The figure indicates forces,velocity, and acceleration at this instant. The total force on the block is now

F = −mgy + N y . (6.19)

Once more the net acceleration is vertical. In order to be moving around a circleat speed vb we must have a = v2

b/Ry directed upward towards the center of thecircle. This requires

N = mg + mv2

b

R. (6.20)

This makes sense. At the bottom, in addition to holding up the block’s weight,the track must apply additional normal force to provide the centripetal acceler-ation.

To find the value of N we again use energy conservation to find vb. At the bottomof the loop the gravitational potential energy vanishes but the conserved totalenergy is equal to its value at the top of the ramp (or at any other time duringthe block’s travels). This means

Eb =mv2

b

2= Ei = mgH , (6.21)


orv2

b = 2gH = 5gR , (6.22)

where the last equality used (6.18). Inserting this we find

N = mg + 5mg = 6mg . (6.23)

If you are sitting in this block and travelling at the minimal speed needed totraverse the loop, then at the top of the loop (where N = 0 you will just barelytouch your seat. At the bottom your seat needs to apply six times your weightto your bottom to accelerate you up. Your diaphragm needs to apply six timesthe force it is accustomed to to hold up the contents of your abdominal cavity,and most importantly your heart must lift your blood out of your feet againstan apparent 6g of gravity. This is why pilots of WWII planes that first achievedhigh speeds had trouble with blacking out. Their hearts failed to overcome theincreased apparent gravity at the bottom of maneuvers and their oxygen-starvedbrains lost consciousness. The remedy at the time was inserting wood blockson the pedals, to raise their feet and put them in a cramped position amenableto tightening their abdominal muscles to restrict blood flow. Modern pressuresuits simply squeeze the lower extremities in any configuration so that bloodflow is unaffected by acceleration.


Problem 106.

problems/wme-pr-loop-the-loop-difficult.tex

H

R

θ

A block of mass M sits at the top of a frictionless hill of height H . It slidesdown and around a loop-the-loop of radius R to an angle θ (an arbitrary one isshown in the figure above, which may or may not be the easiest one to use toanswer this question).

Find the magnitude of the normal force as a function of the angle θ. From this,deduce an expression for the angle θ0 at which the block will leave the track ifthe block is started at a height H = 2R.


Problem 107.

problems/wme-pr-loop-the-loop-from-spring.tex

R

∆x

k

A block of mass M sits in front of a spring with spring constant k compressedby an amount ∆x on a frictionless track leading to a circular loop-the-loop ofradius R as shown.

a) Draw two force diagrams, one with the block at the top of the loop andone with the block at the bottom of the loop. Clearly label all forces, in-cluding ones that you might set to zero or ignore. Use these force diagramsto help answer the following two questions.

b) Find the minimum value of ∆x for which the block barely goes around theloop staying on the track at the top.

c) If the block is started at this position, what is the normal force exertedby the track at the bottom of the loop, where it is greatest?


Problem 108.

problems/wme-pr-loop-the-loop-skier-ambitious-amy.tex

h

m

H

R

Ambitious Amy has a mass m. She skis from initial rest down the (frictionless)ski slope of height H to a ski ramp whose radius of curvature R and whoselowest point is h above the ground (as shown).

Amy’s leg strength must oppose her apparent weight at the bottom of the jump.Is she strong enough? It would be good to know how strong she has to be sothat she can work on leg presses if need be before trying the actual jump. So(in terms of the given quantities m, g, R, H, h) :

a) How fast is Amy going when she reaches the lowest point in the curvedjump?

b) What is the total force that must be directed towards the center of thecircle of motion at that point (again, in terms of the given).

c) Using your knowledge of the actual forces acting on her that have to sumto this force, determine her ”apparent weight” – the peak force she hasto push down on the ground with her skis with in order to stay on thecircular curve.


Problem 109.

problems/wme-pr-loop-the-loop-skier.tex

R

H = 3.5 R

A skier of mass m at an exhibition wants to loop-the-loop on a special (friction-less) ice track of radius R set up as shown. Suppose H = 3.5R. All answersshould be given in terms of g, m and R. (Note that the skier is really muchshorter than R; the picture is not drawn strictly to scale for ease of viewing.)

a) What is her apparent “weight” (the normal force exerted by the track onher skis) when she is upside down at the top of the loop-the-loop?

b) What is her maximum apparent “weight” on the loop-the-loop and where(at what point on the loop-the-loop track) does it occur? Indicate theposition on the figure.


Problem 110.

problems/wme-pr-loop-the-loop-skier-weight.tex

R

H = 3.5 R

A skier of mass m at an exhibition wants to loop-the-loop on a special (friction-less) ice track of radius R set up as shown. Suppose H = 3.5R. All answersshould be given in terms of g, m and R. (Note that the picture is not drawnstrictly to scale for ease of viewing.)

a) What is her apparent weight (the normal force exerted by the track onher skis) when she is upside down at the top of the loop-the-loop? If sheclosed her eyes, what direction would she think of as “down”?

b) What is her maximum apparent “weight” on the loop-the-loop and where(at what point on the loop-the-loop track) does it occur? Indicate theposition on the figure.


Problem 111.

problems/wme-pr-loop-the-loop.tex

H

R

θ


a) Find the minimum height Hmin for which the block barely goes around theloop staying on the track at the top.



Problem 112.

problems/wme-pr-loop-the-loop-to-spring.tex

R

∆x

k+x

+y

xeq

vb

vt

m

A block of mass m is travelling to the right at the top of a frictionless circularloop-the-loop track of radius R, travelling at speed vt to the right as shown. vt

is large enough that the mass remains on the track at the top. It then slidesaround the track to the bottom, slides across the (frictionless) ground, andhits a spring with spring constant k which slows it to rest after the spring hascompressed a distance ∆x from its initial equilibrium length.

a) What is the speed vb at the bottom of the circular loop?

b) What is the normal force exerted by the track at the bottom just before/asit leaves the circular loop?

c) By what distance ∆x is the spring compressed at the instant the blockcomes to rest?


Problem 113.

problems/wme-pr-razor-cuts-loop-string.tex

R

razor

= gRv

A small ball (which can be treated as a point particle for this problem) is at-tached to an Acme (massless, unstretchable) string whose other end is attachedto a fixed, frictionless pivot. The ball swings in a vertical circle, with gravityacting downward as usual.

When the ball is at the top of the circle it has velocity√

gR to the left asshown. After the ball has gone three quarters of the way around and the stringis horizontal, a razor blade cuts the string. You can assume that the impulsedelivered to the string by the razor is small enough that it can be ignored.

a) Find the tension in the string just before it is cut.

b) On the diagram, show the path of the ball after the string is cut. Describein words any features of the path that you intended to illustrate and besure to indicate the maximum height you expect the ball to reach relativeto the center of the circle of motion.


Problem 114.

problems/wme-pr-sliding-off-a-cylinder.tex

m

O

θR

In the figure above, a small (treat as a point mass) block of mass m is on topof a frictionless cylinder so that its center of mass is a distance R from the axisof the cylinder. It is given a nudge so that it slides with negligible initial speeddown the side of the cylinder.

a) When its angular position is θ as shown, what is its speed (assuming thatit is still on the cylinder)?

b) What is the magnitude of the normal force exerted on the block by thecylinder at this point?

c) For what value of θ will the block leave the cylinder?

Express your answers in m, R, g and θ.


Problem 115.

problems/wme-pr-smooth-inclined-plane-friction-table-2.tex

µθ m0

k v

D

y

x

A block of mass m is given a push so that it slides at a speed v0 from the right tothe left over a smooth (frictionless) surface, until it hits a patch of rough surfaceof length D leading up to a smooth (frictionless) incline. The incline makes anangle θ with the horizontal as shown. The coefficient of friction between theblock and the rough surface is µk.

a) What is the minimum speed v0,min such that the block will reach thebottom of the incline?

b) Assuming that the block is travelling at some v0 > v0,min, how high (towhat maximum height ymax) will the block slide up the incline (use thecoordinate system given)?


Problem 116.

problems/wme-pr-smooth-inclined-plane-friction-table.tex

L

m

µk

D?

at rest

θ

A block of mass m slides down a smooth (frictionless) incline of length L thatmakes an angle θ with the horizontal as shown. It then reaches a rough surfacewith a coefficient of kinetic friction µk.

a) How fast is the block going as it reaches the bottom of the incline?

b) What distance D down the rough surface does the block slide before com-ing to rest?


Problem 117.

problems/wme-pr-spring-to-inclined-plane-friction-numbers.tex

θµk,s

frictionless

v?x

keq

with friction

L?m

x0

A block of mass M = 1 kg is propelled by a spring with spring constant k = 10N/m onto a smooth (frictionless) track. The spring is initially compressed adistance of 0.5m from its equilibrium configuration (xi − x0 = 0.5 m). At theend of the track there is a rough inclined plane at an angle of 45 with respectto the horizontal and with a coefficient of kinetic friction µk = 0.5.

a) How far up the incline will the block slide before coming to rest (find Hf )?

b) The coefficient of static friction is µs = 0.7. Will the block remain at reston the incline? If not, how fast will it be going when it reaches the bottomagain?


Problem 118.

problems/wme-pr-spring-to-inclined-plane-friction.tex

θµk,s

frictionless

v?x

keq

with friction

L?m

x0

A block of mass M kg is propelled by a spring with spring constant k N/monto a smooth (frictionless) track. The spring is initially compressed a distanceof x0 from its equilibrium configuration. At the end of the track there is arough inclined plane at an angle of θ with respect to the horizontal and with acoefficient of kinetic friction µk.

a) How far up the incline will the block slide before coming momentarily torest (find Lmax)?

b) Suppose the coefficient of static friction is µs. Find the maximum angleθrmmax such that the block will remain at rest at the top of the inclineinstead of sliding back down.


Problem 119.

problems/wme-pr-spring-to-plane-friction.tex

∆x

kµ

k v?

with kinetic friction

m

frictionless

xeq

D

A block of mass m sits against a spring with spring constant k that is initiallycompressed a distance ∆x. At some time the block is released and slides acrossa frictionless surface until it reaches a rough surface with a coefficient of kineticfriction µk as shown.

a) How fast is the block going as it leaves the spring at xeq?

b) What distance D down the rough surface does the block slide before com-ing to rest?


6.3 Power


Problem 120.

problems/power-mc-two-blocks.tex

F2m

Fm

B

A

In the figures A and B above an identical force is applied to two masses sittinginitially at rest on a frictionless table. In both cases the force is applied forthe same amount of time. Identify the true statement in the list below.

a) The power provided to both blocks is identical throughout this time, andthey end up with the same final kinetic energy.

b) The power provided to the first block A is larger than that applied to thesecond, and A ends up with more kinetic energy.

c) The power provided to the second block B is larger than that applied toblock A, but block A travels further.

d) It is impossible to tell from the information given which block receivesmore power from the force.

6.3. POWER 151

6.3.2 Long Problems

Problem 121.

problems/power-pr-constant-power-v-of-t.tex

F

v

P0

m

In the figure above, a mass m is pulled along on a frictionless table by a motorwith constant power – it does work on whatever is attached to its drivewheelat rate P0. At the instant shown, the mass has reached a speed v towards themotor.

a) Find F as a function of P0 and v (in the direction of the motor).

b) Write Newton’s second law for the mass m in terms of your answer to a),using a = dv/dt for the acceleration.

c) Solve the equation of motion you get for v(t), assuming that v(0) = 0.

d) Qualitatively sketch what you expect to get for v(t) (or what you did getin the previous section). Note that you can do this one even if you fail todo the integral correctly, if you think about what happens to the force asthe speed gets bigger and bigger.

Chapter 7

Center of Mass andMomentum

With basic dynamics and kinetics (force and energy) under our belts for massiveparticles taken one or two at a time (or even three or four at a time) we needto move on and see what happens when we treat lots of particles at a time –arbitrarily many. After all, eventually we want to understand solids that canrotate as well as translate, and fluids that, well, act like fluids. Neither one canbe thought of precisely like a particle.

Or can they? As long as we consider a system of particles to be a single particle”located at its center of mass, nearly everything we’ve done so far can be appliedto the entire system, even if the system is a liquid or a gas with many, manyparticles and no particular structure!

When do we want to use the concepts of center of mass and momentum conser-vation? Momentum conservation works for isolated systems (or not-so-isolatedsystems in the impulse approximation) with no net external force acting on it,especially to understand collisions. Center of mass is a useful concept boththen and when a collection of particles is being acted on in a uniform way byan external force (such as near-Earth gravity).

153

154 CHAPTER 7. CENTER OF MASS AND MOMENTUM

7.1 Center of Mass


Problem 122.

problems/center-of-mass-mc-particles-1.tex

E

A

C

m

m m

m

D B

A collection of four equal masses m (including the uniform half-circle of wire)is shown above. Which of the points A-E is a plausible location of the center ofmass?

7.1. CENTER OF MASS 155

Problem 123.


Bm

m

m

m

AD

C

Pick (circle) the point A-D most likely to be the center of mass of the systemabove, given four equal masses m arranged as shown. Note that the dashed lineis drawn (connecting the mass centers) simply as a guide to the eye.


Problem 124.


y

x2 kg 3 kg1m 2m

2m

1m

2 kg1.5 m

1 kg

In the figure above, various given masses (in kilograms) are located at the po-sitions shown shown. The center of mass of this system is at:

a) x = 5/4m, y = 1/2m

b) x = 1m, y = 1/2m

c) x = 1/2m, y = 1/4m

d) x = 3/4m, y = 1m

e) x = 1/2m, y = 1m


Problem 125.

problems/center-of-mass-mc-particles-4.texy

x2 kg1m 2m

2m

1m

2 kg1.5 m

3 kg

1 kg

In the figure above, various given masses (in kilograms) are located at the po-sitions shown shown. The center of mass of this system is at:

a) x = 5/4m, y = 1/2m

b) x = 3/4m, y = 5/8m

c) x = 1/2m, y = 1/4m

d) x = 1m, y = 1/2m

e) x = 5/8m, y = 3/4m


Problem 126.


y

x2 kg 3 kg1m 2m

1m

1 kg2m

2 kg

In the figure above, various given masses (in kilograms) are located at the cornersof a square with sides of length 2 meters as shown. The center of mass of thissystem is at:

a) x = 5/4m, y = 1/2m

b) x = 1m, y = 1/2m

c) x = 3/2m, y = 3/4m

d) x = 5/4m, y = 3/4m

e) x = 3/2m, y = 1/2m


Problem 127.


y

x2 kg 3 kg1m 2m

1m

2m

2 kg 1 kg

In the figure above, various given masses (in kilograms) are located at the po-sitions shown above. The center of mass of this system is at:

a) x = 5/4 m, y = 3/4 m

b) x = 1 m, y = 3/4 m

c) x = 1 m, y = 1 m

d) x = 5/3 m, y = 5/4 m

e) x = 3/4 m, y = 1 m


7.1.2 Short Answer

Problem 128.

problems/center-of-mass-sa-particles.tex

figure In the figure below, various given masses (in kilograms) are located atthe corners of a square with sides of length 10 meters as shown. Fill in thecoordinates of the center of mass of this system below and place an “x” on thethe graph at its location.

1 kg 4 kg

3 kg2 kg

−5m

5m

x

y

−5m 5m

xCoM =

yCoM =


7.1.3 Long Problems

Problem 129.

problems/center-of-mass-pr-arc-270-example.tex

x

y

M

R

In the figure above, a uniformly thick piece of wire is bent into 3/4 of a circulararc as shown. Find the center of mass of the wire in the coordinate systemgiven, using integration to find the xcm and ycm components separately.


Problem 130.

problems/center-of-mass-pr-arc-270-solution.tex

x

y

M

R

This problem will help you learn required concepts such as:

• Center of Mass

• Integrating a Distribution of Mass

so please review them before you begin.

In the figure above, a uniformly thick piece of wire is bent into 3/4 of a circulararc as shown. Find the center of mass of the wire in the coordinate systemgiven, using integration to find the xcm and ycm components separately.

The uniform wire can be assumed to have a uniform mass per unit length µ.We will assume that the wire’s thickness is far smaller than R so to computethe center of mass we can take the entire mass of the wire to lie along one circle(neglecting the different positions of various parts of the wire’s cross-section).

We can then parameterize points on the wire using the angle θ from the positivex-axis as shown, just as we did for motion along a circle. The coordinates ofpoints on the circle are, in Cartesian coordinates,

r = R cos(θ)x + R sin(θ)y , (7.1)

and the wire extends over the range 0 ≤ θ ≤ 3π/2. The length of the segmentof wire represented by the angular interval from θ to θ+dθ is µ times the lengthRdθ of the interval, i.e.

dM = Rµdθ . (7.2)


The total mass of the wire is thus

M =

∫

dM =

∫ 3π2

0

Rµdθ =3π

2µR . (7.3)

The center of mass position is then

rCOM = (1/M)

∫

rdm

= (1/M)

∫ 3π/2

0

(R cos(θ)x + R sin(θ)y)Rµdθ

=R2µ

M

∫ 3π/2

0

(cos(θ)x + sin(θ)y) dθ

=2R

3π(−x + y) .


Problem 131.

problems/center-of-mass-pr-circular-cone.tex

θ0

z

x

y

H

Above is drawn a circular cone with uniform mass density ρ. The cone sidemakes an angle θ0 with the positive z axis. The cone height is H . Find thecenter of mass of the cone in terms of the quantities given above. Hint: Considercircular slabs of thickness dz a height z above the origin.


Problem 132.

problems/center-of-mass-pr-dog-in-a-boat.tex

LL/2

M

LD

M m

m

A dog of mass m is sitting at one end of a boat of mass M and length L thatis sitting next to a dock as shown. The dog decides he wants some tasty dogchunks that are waiting for him at home and walks to the other end of the boat,expecting to step out onto the dock. Sadly, when he gets there he finds himselfa distance D away from the dock.

a) What is D in terms of m, M , and L. You may assume that the boat issymmetric, so that its center of mass is at L/2, although this is not strictlynecessary to get the answer.

b) The dog thinks he can jump for it if D < L/2, but otherwise he’ll have toswim. Find the ratio m/M for which the dog has to take a bath.


Problem 133.

problems/center-of-mass-pr-right-triangle.tex

x

M

y

a

by(x) = (b/a)x

In the figure above, a uniformly thick sheet of plastic is cut into the a× b righttriangle shown. Find the center of mass of the triangle in the coordinate systemgiven, using integration to find the xcm and ycm components separately.

Suggested solution strategy:

a) Form σ = M/A where A is the area of the triangle.

b) Form dm = σ dA where dA = dx dy.

c) Do the integrals∫

x dm and∫

y dm separately, using the provided func-tional form of the hypotenuse to set up the limits of integration in bothcases.

d) Divide out the M to obtain xcm and ycm.


Problem 134.

problems/center-of-mass-pr-rod-variable-lambda.tex

0 L +x

2M2L

λ (x) = x


• Center of Mass of Continuous Mass Distributions

• Integrating Over Distributions


In the figure above a rod of total mass M and length L is portrayed that hasbeen machined so that it has a mass per unit length that increases linearly alongthe length of the rod:

λ(x) =2M

L2x

This might be viewed as a very crude model for the way mass is distributed insomething like a baseball bat or tennis racket, with most of the mass near oneend of a long object and very little near the other (and a continuum in between).

Treat the rod as if it is really one dimensional (we know that the center of masswill be in the center of the rod as far as y or z are concerned, but the rod is sothin that we can imagine that y ≈ z ≈ 0) and:

a) verify that the total mass of the rod is indeed M for this mass distribution;

b) find xcm, the x-coordinate of the center of mass of the rod.


Problem 135.

problems/center-of-mass-pr-romeo-and-juliet.tex

v0

mw

L

M b

Romeo and Juliet are out in their damn boat again, this time for a picnic onthe lake. The boat is initially at rest. Juliet decides she wants a piece oftasty watermelon, and throws the watermelon at horizontal speed v0 to Romeoat the other end of the boat a distance L away so he can cut her a piece withhis ever-handy bodkin (dagger). The combined mass of Romeo, Juliet and theboat is Mb; the mass of the watermelon is mw. Assume that the boat can movehorizontally on the water without drag or friction.

a) What is the horizontal speed of the boat while the watermelon is in theair (neglect its vertical motion – assume that Juliet has thrown it on a flattrajectory as shown).

b) What is the horizontal speed of the boat after Romeo catches the water-melon?

c) How long is the watermelon in the air?


Problem 136.

problems/center-of-mass-pr-semicircular-sheet.tex

x

y

R

Find the center of mass of the two-dimensional semicircular sheet drawn above.It has a uniform mass per unit area σ and radius R. You may invoke symmetryfor one of the two vector components of the center of mass location.


7.2 Momentum


Problem 137.

problems/momentum-mc-also-energy-football.tex

Two football players, one large (L – bigger mass) and one small (S – smallermass) are running in a straight line directly at one another. They have the samemagnitude of momentum. Rank their mechanical energies and speeds rightbefore they collide.

a) ES < EL, vS < vL

b) ES < EL, vS > vL

c) ES > EL, vS > vL

d) ES > EL, vS < vL

7.2. MOMENTUM 171

Problem 138.

problems/momentum-mc-baseball-and-bat.tex

Michelle is playing baseball and hits a home run with a solid wood bat (massof 3 kg). The baseball (mass of 0.5 kg) is knocked clean out of the park. Theforce exerted by the bat on the baseball is:

a) Greater (in magnitude) than the force exerted by the baseball on the bat;

b) Less than the force exerted by the baseball on the bat;

c) The same as the force exerted by the baseball on the bat.


Problem 139.

problems/momentum-mc-cement-truck-and-bug.tex

A cement truck with a mass Mt is travelling at speed vt collides with a bug ofmass mb that is hovering above the road (so vb ≈ 0). One can safely assume thatMt ≫ mb. Which of the following statements are unambiguously true (circle alldefinitely true statements)?

a) If the bug recoils off of the windshield elastically, its final speed is roughly2vt (in the same direction as the truck).

b) The magnitude of the momentum change of the truck is much smallerthan the magnitude of the momentum change of the bug.

c) If the bug splatters and sticks to the windshield of the truck, the totalkinetic energy of the bug and truck will be conserved.

d) At all times during the collision, the bug exerts exactly the same magni-tude of force on the truck that the truck exerts on the bug.

e) The final speed of the bug as it recoils off of the windshield is roughlyMt

mbvt (in the same direction as the truck).

7.2. MOMENTUM 173

Problem 140.

problems/momentum-mc-elastic-collision-blocks.tex

2mv mv

+xv

m2m

before after

2m m

A block of mass 2m slides on a frictionless table at velocity ~v = vx to theright (positive x-direction) to collide with a block of mass m initially at restas shown. Assuming that the collision is one dimensional and elastic, thevelocities of the two blocks after the collision are:

a) ~v2m = v2 x ~vm = 3v

2 x

b) ~v2m = 0x ~vm =√

2vx

c) ~v2m = − v3 x ~vm = 4v

3 x

d) ~v2m = v3 x ~vm = 4v

3 x


Problem 141.

problems/momentum-mc-fission-1.tex

0v

3mv0

v?

m x

y

2m

An atomic nucleus of mass 3m is travelling to the right at velocity ~vinitial = v0i

as shown. It spontaneously fissions into two fragments of mass m and 2m.The smaller fragment m travels straight down at velocity ~vm = −v0j after thefission. What is the velocity of the larger fragment?

a) ~v2m = 32v0i

b) ~v2m = 2v0j

c) ~v2m = 32v0i + 1

2v0j

d) ~v2m = − 32v0i − 1

2v0j

e) ~v2m = 3v0i + 2v0j

7.2. MOMENTUM 175

Problem 142.

problems/momentum-mc-fission-2.tex

x

y

4m, v4m = 2v0

8m, v8m

12m, v0

= ?

An atomic nucleus of mass 12m is travelling to the right at velocity ~vinitial = v0x

as shown. It spontaneously fissions into two fragments of mass 4m and 8m(releasing energy). The smaller fragment 4m travels straight down at velocity~v4m = −2v0y after the fission. What is the velocity of the larger fragment?

a) ~v8m = 12v0x + v0y

b) ~v8m = 2v0x + 2v0y

c) ~v8m = − 32v0x − v0y

d) ~v8m = 32v0x + v0y

e) ~v8m = 32v0x + 2v0y


Problem 143.

problems/momentum-mc-simple-inelastic-collision.tex

m 3m0v

0−v

A mass m travelling at (one-dimensional) velocity v0 to the right collides withmass 3m travelling at velocity −v0 to the left and sticks to it. The final velocityvf of the blocks after the collision is:

a) vf = −2v0

b) vf = v0/2

c) vf = −v0

d) vf = −2v0/3

e) vf = −v0/2

7.2. MOMENTUM 177

Problem 144.

problems/momentum-mc-two-masses-spring.tex

m1 m2

Two masses, m2 = 2m1 are separated by a compressed spring. At time t = 0they are released from rest and the (massless) spring expands. There is nogravity or friction. As they move apart, which statement about the magnitudeof each mass’s kinetic energy Ki and momentum pi is true?

a) K1 = K2, p1 = 2p2

b) K1 = 2K2, p1 = p2

c) K1 = K2/2, p1 = p2/2

d) K1 = K2, p1 = 2p2

e) K1 = K2/4, p1 = p2


Problem 145.

problems/momentum-mc-two-trucks-collide.tex

A fully laden dump truck (mass of maybe 10,000 kg) slams into a small pickuptruck (mass around 2,000 kg). The two trucks exert a collision force on oneanother, and momentum is transferred during the short collision.

Let FD, ∆pD, aD represent the magnitude of the force exerted by the dumptruck on the pickup truck, the magnitude of the dump truck’s momentumchange, and the magnitude of the dump truck’s average acceleration duringthe collision. Let Fp, ∆pp, ap represent the magnitude of the force exerted bythe pickup truck on the dump truck, the magnitude of the pickup truck’s mo-mentum change, and the magnitude of the pickup truck’s average accelerationduring the collision.

Select the correct/true description of these magnitudes below:

a) FD = Fp, ∆PD > ∆Pt, aD < at

b) FD > Fp, ∆PD = ∆Pt, aD < at

c) FD = Fp, ∆PD > ∆Pt, aD > at

d) FD = Fp, ∆PD = ∆Pt, aD < at

e) FD = Fp, ∆PD = ∆Pt, aD = at

f) None of the above (enter the correct answer here):

7.2. MOMENTUM 179

7.2.2 Short Answer

Problem 146.

problems/momentum-sa-elastic-recoil.tex

(incident) (target)

v

vm m

m

mM

M

o

vo

o(a)

(b)

(c)

In the three figures above, mass M > m. The mass on the left is incident atspeed v0 on the target mass (initially at rest in all three cases) on the right .The two particles undergo an elastic collision in one dimension and the targetmass recoils to the right in all three cases. In the spaces provided below youare asked to provide a qualitative estimate of the speed and direction of theincident particle after the collision.

Your answer should be given relative to v0 and should look like “vx > v0, tothe left” or “vx = 0” or vx < v0, to the right” where x = a, b, c. In otherwords, specify the speed qualitatively compared to v0 and then the direction,per figure.

a)

b)


c)

7.2. MOMENTUM 181

Problem 147.

problems/momentum-sa-hammer-impacts-head.tex

m

H

A hammer of mass m falls from rest off of a roof and drops a height H ontoyour head. Ouch!

a) Assuming that the tool is in actual contact with your head for a time ∆tbefore it stops (thud!) and slides off, what is the algebraic expression forthe average force it exerts on your hapless skull while stopping?

b) Estimate the magnitude of this force using m = 1 kg, H = 1.25 meters,∆t = 10−2 seconds and g = 10 m

sec2 . Compare this force to the weight ofthe hammer of 10 Newtons!


Problem 148.

problems/momentum-sa-impact-orc-spears-frodo.tex

An Orc throws a 2 kg spear at Frodo Baggins at point blank range, but it isstopped by his hidden mithril mail shirt. Assuming that the spear was travellingat 20 m/sec when it hit and that it stopped in 0.1 seconds, what was the averageforce exerted on the spear by the mail coat (and the hobbit underneath)? Ouch!

7.2. MOMENTUM 183

Problem 149.

problems/momentum-sa-shark-eats-fish.tex

A great white shark of mass m1, coasting through the water in a nearly fric-tionless way at speed v1, engulfs a tuna of mass m2 < m1 travelling in the samedirection at speed v2 < v1, swallowing it in one bite.

a) What is the speed of the shark after its tasty meal, sadly eaten withoutwasabi (mmm, sashimi!)?

b) Did the shark gain (kinetic) energy, lose energy, or have its energy remainthe same in the process.


7.2.3 Long Problems

Problem 150.

problems/momentum-pr-ballistic-pendulum-partially-inelastic.tex

m,v m,v

L

0 1

M,v = 0

M,v = 0f

θmax

In the figure above, a bullet of mass m and initial velocity v0 passes througha block of mass M suspended by an unstretchable, massless string of length Lfrom an overhead support as shown. It emerges from the collision on the farside travelling at v1 < v0. This happens extremely quickly (before the block hastime to swing up) and the mass of the block is unchanged by the passage of thebullet (the mass removed making the hole is negligible, in other words). Afterthe collision, the block swings up to a maximum angle θmax and then stops.

Find θmax.

7.2. MOMENTUM 185

Problem 151.

problems/momentum-pr-bullet-block-free-fall-1.tex

H

R

vm M0

A bullet of mass m travelling at speed v0 in the direction shown above strikesa block of mass M and embeds itself in it. The block is sitting on the edge of africtionless table of height H and is knocked off of the table by the collision.

a) What is the speed vb of the block immediately after the bullet sticks?

b) What distance R from the base of the table does the block land?

Note: If you cannot solve a), just use the symbol vb where needed to get possiblyfull credit for b). Do not just use a memorized formula for b): Clearly state thephysical principle(s) you are using and work out the answers.


Problem 152.

problems/momentum-pr-bullet-block-free-fall-2.tex

H

R

v1 v2m M

A bullet of mass m travelling at speed v1 in the direction shown above strikes ablock of mass M and passes through, emerging at speed v2. The block is sittingon the edge of a frictionless table of height H and is knocked off of the table bythe collision.

a) What is the speed vb of the block immediately after the bullet emerges?

b) What distance R from the base of the table does the block land?

Note: If you cannot solve a), just use the symbol vb where needed to get possiblyfull credit for b). Do not just use a memorized formula for b): Clearly state thephysical principle(s) you are using and work out the answers.

7.2. MOMENTUM 187

Problem 153.

problems/momentum-pr-bullet-stops-at-block.tex

M

m

vb

vi

(bullet stops)

bv

In the figure above a bullet of mass m is travelling at initial speed v0 to the rightwhen it strikes a larger block of mass M that is resting on a rough horizontaltable (with coefficient of friction between block and table of µk).

Instead of “sticking” in the block, the bullet is stopped cold by the block andfalls to the ground, while the block recoils from the collision to the right. Notethat this collision is partially inelastic, so some mechanical energy will be lost.

a) What is the velocity of the block vb immediately after the collision.

b) How much energy is lost in the collision?

c) Find the distance D that the block slides down the table before comingto rest after the collision.


Problem 154.

problems/momentum-pr-bullet-through-block-rough-surface.tex

M

m

D

µ

f

vb

v

vik

(block at rest)


• Momentum Conservation

• The Impact Approximation

• Elastic versus Inelastic Collisions

• The Non-conservative Work-Mechanical Energy Theorem


In the figure above a bullet of mass m is travelling at initial speed vi to the rightwhen it strikes a larger block of mass M that is resting on a rough horizontaltable (with coefficient of friction between block and table of µk). Instead of“sticking” in the block, the bullet blasts its way through the block (withoutchanging the mass of the block significantly in the process). It emerges with thesmaller speed vf , still to the right.

a) Find the speed of the block vb immediately after the collision (but beforethe block has had time to slide any significant distance on the roughsurface).

b) Find the (kinetic) energy lost during this collision. Where did this energygo?

7.2. MOMENTUM 189

c) How far down the rough surface D does the block slide before coming torest?


Problem 155.

problems/momentum-pr-bullet-through-block.tex

M

m

f

vb

v

vi

In the figure above a bullet of mass m is travelling at initial speed vi to theright when it strikes a larger block of mass M that is resting on a horizontaltable. Instead of “sticking” in the block, the bullet blasts its way through theblock (without changing the mass of the block significantly in the process). Itemerges with the smaller speed vf , still to the right.

a) What is the velocity of the block vb immediately after the collision.

b) How much energy is lost in the collision?

7.2. MOMENTUM 191

Problem 156.

problems/momentum-pr-dog-jumps-from-boat.tex

m

vb

dv

M D

A dog of mass m gets hungry while sitting at the end of a boat of mass M andlength L that is at rest on the water of a lake. He jumps out onto the dock togo get some tasty dog chunks that are waiting for him at home when the boatis a distance D away from the dock as shown. The dog travels at a horizontalspeed vd relative to the ground/lake as he flies through the air.

a) What is the recoil speed of the boat, vb, while the dog is in the air? Assumethat dog and boat are both at rest before the jump.

b) How much work did the dog’s legs do during the jump?


Problem 157.

problems/momentum-pr-elastic-collision-proton-neon.tex

mp p

vo

M = 20m

A Proton of mass mp is directly incident on a Neon nucleus with mass 20mp. It isinitially (far away from the nucleus) travelling with speed v0. The two particlesrepel each other (like charges repel) as they approach, and the force of repulsionis strong enough to prevent the particles from touching. The “collision” thattakes place gradually between the two particles is elastic.

a) At some distant time in the future (after the collision) is the proton movingto the left or to the right?

b) What is the speed of the proton when it and the Neon nucleus are at thedistance of closest approach?

c) What is the speed of the Neon nucleus at a distant time in the future(after the collision) when they are once again far apart.

7.2. MOMENTUM 193

Problem 158.

problems/momentum-pr-elastic-two-balls.tex

v

m

m

v

vo

t

b

θθb

t

In the figure above, a ball with mass m = 1kg and speed v0 = 5 m/sec elas-tically collides with a stationary, identical ball (all resting on a frictionlesssurface so gravity is irrelevant). A student measures the top ball emerging fromthe collision at a speed vt = 4 m/sec at an angle θt ≈ 37 as shown.

a) Find the speed vb of the other ball.

b) Find the angle θb of the other ball. (Hint: Draw a triangle with sides oflength v0, vt, vb.)

c) What does θt+θb add up to? (This is a characteristic of all elastic collisionsbetween identical masses in 2 dimensions.)


Problem 159.

problems/momentum-pr-fission.tex

0v

3mv0

v?

m x

y

2m

An atomic nucleus of mass 3m is travelling to the right at velocity ~vinitial = v0x

as shown. It spontaneously fissions into two fragments of mass m and 2m.The smaller fragment m travels straight down at velocity ~vm = −v0y after thefission.

a) What is the velocity of the larger fragment?

b) What is the net energy gain or loss (indicate which!) from the fissionprocess, in terms of the initial kinetic energy?

7.2. MOMENTUM 195

Problem 160.

problems/momentum-pr-inelastic-collision-ball-bearing-spring.tex

x

m

M

v

block comes to rest

collision

A ball bearing of mass m = 50 grams travelling at 200 m/sec smacks into ablock of mass M = 950 gms and sticks in a hole drilled therein. The block isinitially at rest on an Acme frictionless table and is also connected to an Acmespring with spring constant k = 400 N/m at its equilibrium position (see figure).

a) What is the maximum distance x the spring is compressed by the recoilingball bearing-block system?

b) How much mechanical energy is lost in the collision (noting that an answerof ‘none’ is one possibility)?


Problem 161.

problems/momentum-pr-inelastic-elastic-collision.tex

m2m

3m

kv

A B

Two masses A and B rest on a frictionless surface, with a massless spring withspring constant k connected to B. A bullet coming from the left with speed vhits A and becomes embedded in it. The masses of the bullet, A and B are m,2m and 3m respectively.

a) What is the speed vcm of the center of mass of the system consisting of A,B and the bullet?

b) Immediately after A gets hit by the bullet, what is the speed vA of A (withthe bullet embedded in it) before it hits the spring?

c) In the subsequent motion of the system, what is the maximum compression∆xmax of the spring?

7.2. MOMENTUM 197

Problem 162.

problems/momentum-pr-inelastic-football-collision.tex

v0

v0/2M

3M/2

θ

x

y

A running back of mass M is running at speed v0 upfield. A linebacker of mass3M/2 is running along one of the yardlines (so his velocity is at a right angle tothe running back’s) at a speed 1

2v0. The linebacker tackles the running backin mid-air so that the two bodies stick together. Answer the questions belowin terms of the givens above and show your reasoning.

a) By what angle θ is the running back deflected from his original direction?

b) What is the horizontal velocity of the two right after the collision in thex-y plane while they are still in the air?

c) Who experiences the greatest magnitude of acceleration during thecollision, the running back (M) or linebacker (3M/2)?


Problem 163.

problems/momentum-pr-neutron-collides-with-helium.tex

n0v

initially at rest

Hex

y

In the figure above, a neutron of mass m collides elastically with a heliumnucleus of mass 4m, striking it head on so that the collision is one dimensional.The initial speed of the the neutron is v0; the helium nucleus is initially at rest.In answering the following questions you may either find or just remember thesolution for one dimensional elastic collisions – you do not have to derive it,although you may if you wish or cannot remember it.

a) What is the final velocity of the neutron (magnitude and direction) afterthe collision.

b) What is the final velocity of the helium nucleus after the collision.

c) Is the helium nucleus moving faster or slower than the neutron is movingafter the collision? (Does your answer make sense?)

7.2. MOMENTUM 199

Problem 164.

problems/momentum-pr-two-astronauts.tex

M m M

L

v0

Two astronauts with identical mass M (including their spacesuits) in free-fallare working on a satellite. They are connected by a taut tether rope of lengthL. The first astronaut on the needs a tool of mass m that the second astronautis carrying (also initially a distance L away), so the second one tosses the toolto the first at speed v0.

a) What is the speed of the two astronauts while the tool is in space flyingfreely between them?

b) What is the speed of the two astronauts after the first one catches thetool?

c) The first astronaut cannot reach the satellite if he has drifted a distanced further away while the tool was in flight. Find the maximum length Lthat the tether can have such that the first astronaut can still reach thesatellite. Express your answer in terms of M , m, v0 and d as needed.

Chapter 8

One Dimensional Rotationand Torque

OK, so whole systems behave like particles, as long as the particle” is located atthe center of mass of the system of particles. But solid objects have a secondkind of motion. They can rotate around an axis through their center of mass(or possibly around other axes if they are suitably constrained).

A bit of algebra and thought transforms Newton’s Second Law and the notionof kinetic energy into rotational forms involving rotations around a single axis,although that axis is itself arbitrary so that we know that eventually we’ll haveto make this a vector theory. For the moment, though, we’ll just add a singleplane-polar angle to the regular coordinate description of the center of mass ofobjects that rotate, or translate and rotate.

You can understand rotation in terms of the simple 1-D stuff you learned the firstweek as a analogous system by thinking of torque as the rotational equivalent offorce, moment of inertia the rotational equivalent of mass, the angle of rotationθ the equivalent of a spatial coordinate like x, and so on.

Don’t forget the rolling constraint used in many of the problems! Also don’tforget to choose your coordinate system – for translation and for rotation –consistently so that angular acceleration can be related to translational accel-eration (and so on) without a spurious minus sign.

Torque is a bit twisted (as physics subjects go), sorry...

201

202 CHAPTER 8. ONE DIMENSIONAL ROTATION AND TORQUE

8.1 Rotation


Problem 165.

problems/rotation-mc-cable-spool-rolls-on-line.tex

Rr

F

A cable spool of mass M , radius R and moment of inertia I = βMR2 around anaxis through its center of mass is wrapped around its outer disk with fishingline and set on a rough rope as shown. The fishing line is then pulled with aforce of magnitude F to the right as shown so that it rolls down the rope on thespool at radius r to the right without slipping.

What is the direction of static friction as it rolls?

a) To the right.

b) To the left.

c) Not enough information to tell (depends on e.g. the size of r relative toR, the numerical value of β, or other unspecified data).

8.1. ROTATION 203

Problem 166.

problems/rotation-mc-rolling-disk-friction-1.tex

F

In the figure above a force is applied to the center of a disk (initially at rest)sitting on a rough table by means of a rope attached to its frictionless axle inthe direction shown. The disk then accelerates and rolls without slipping.The net horizontal force exerted by the table on the disk is:

a) kinetic friction to the right.

b) kinetic friction to the left.

c) static friction to the right.

d) static friction to the left.



Problem 167.

problems/rotation-mc-rolling-disk-friction.tex

F

In the figure above a force is applied to the center of a disk (initially at rest)sitting on a rough table by means of a rope attached to its frictionless axle inthe direction shown. The disk then accelerates and rolls without slipping.The net horizontal force exerted by the table on the disk is:

a) kinetic friction to the right.

b) kinetic friction to the left.

c) static friction to the right.

d) static friction to the left.


8.1. ROTATION 205

Problem 168.

problems/rotation-mc-rolling-race-1.tex

Mr. Hoop Ms. Disk

Mr. Hoop and Ms. Disk had a race rolling down two identical hills withoutslipping. They both started at the top at the same time. Who won?

a) Mr. Hoop

b) Ms. Disk



Problem 169.

problems/rotation-ra-hoop-and-disk.tex

A hoop and a disk of identical mass and radius are rolled up two identicalinclined planes without slipping and reach a maximum height of Hhoop andHdisk respectively before coming momentarily to rest and rolling back down.

Use one of the three signs <, > or = in the boxes below to correctly completeeach statement.

a) If both hoop and disk start with the same total kinetic energy then:

Hhoop Hdisk

b) If both hoop and disk start with the same total center of mass speedthen:

Hhoop Hdisk

c) If both hoop and disk start with the same total center of mass speedthen comparing the magnitude of the work done by gravity :

Wgravity,hoop Wgravity,disk

8.1. ROTATION 207

Problem 170.

problems/rotation-ra-loop-the-loops-balls.tex

r

rr

r

vv

v v

(c)

(b)(a)

(d)

a b

c dstring rod

slidingrolling

In the figures above, (a) shows a ball rolling without slipping on a track; (b)shows the ball sliding on a frictionless track; (c) shows the ball on a string; (d)shows the ball attached to a rigid massless rod attached to a frictionless pivot.In all four figures the ball has the smallest velocity at the bottom of its circulartrajectory that will suffice for the ball to reach the top while still moving in acircle (note that the velocities are not drawn to scale).

Correctly ordinally rank these minimum velocities, for example va = vb < vc <vd is a possible (but probably incorrect) answer.

Note: You must either justify your answer with simple physical arguments or

just solve for the minimum velocity needed at the bottom in terms of m, r,β = 2/5 (for a ball), g and then order the results. You can’t just put downa “guess” for an order with no valid physical reasoning backing it and have itcount, but it is possible to reason your way all or most of the way to an answerwithout doing all of the algebra.


8.1.3 Short Answer

Problem 171.

problems/rotation-sa-rolling-race-1.tex

H

H

H

m

M

m

M

m

m

(c)

(a)

(b)

Its a race! Three wheels made out of two concentric rings of mass connectedby light spokes are identically placed at the top of an inclined plane of heightH as shown. At time t = 0 they are all three released from rest to roll withoutslipping down the incline. You are given the following information about eachdouble ring:

a) Inner ring mass m is less than outer ring mass M .

b) Inner ring mass m is the same as outer ring mass m.

c) Inner ring mass M is greater than outer ring mass m.

In what a, b, c order do the rings arrive at the bottom of the incline? (4 points)

8.1. ROTATION 209

Problem 172.


Hh

M,Rring disk

You are given two inclined planes with different maximum heights H > h asshown. If a ring and disk of identical radius R and mass M are each placed atthe top of one of the two planes and released, they will roll without slipping toarrive at the bottom travelling at the same speed. If placed at the top of theplanes in the other order, they will not.

Draw and label the ring and disk at the tops of the correct planes such thatthey will roll to the bottom and arrive travelling at the same speed.


Problem 173.


ring diskM,R

HH

You are given two inclined planes with the same height H as shown. A ringand disk of identical radius R and mass M are each placed at the top of one ofthe two planes and released at the same instant to roll without slipping to thebottom of their respective inclines..

a) Which one gets to the bottom first? (Circle) ring disk

b) Which one has the greatest speed at the bottom? ring disk

c) Which one has the greatest rotational kinetic energy at the bottom?ring disk

8.1. ROTATION 211

8.1.4 Long Problems

Problem 174.

problems/rotation-pr-asymmetric-atwoods-2.tex

m

M

m

R/2

R

2

1

A pulley of mass M , radius R and moment of inertia I = βMR2 has a mass-less, unstretchable string wrapped around it many times and has a mass m1

suspended from the string. A second massless, unstretchable string is wrappedthe opposite way around a massless, frictionless axle with radius R/2 as shownand has mass m2 suspended from the string. The system begins at rest.

a) What must m2 be in terms of m1 for the system to remain stationary?

b) Suppose m1 = m2 = M . Find ~α, the angular acceleration of the pulleyabout its center of mass. This is a vector! Indicate the direction of theangular acceleration on the figure or in your answer.


Problem 175.

problems/rotation-pr-asymmetric-atwoods.tex

R

m1

m2

M

r

Two objects m1 and m2 (with m1 > m2) are attached to massless unstretchableropes that are attached to wheels on a common frictionless axle as shown in thefigure. The total moment of inertia of the two wheels is given as I. The radiiof the wheels are R (for mass m1 and r (for mass m2 as shown.

a) Show all forces on the two masses and the system of wheels using free-body diagrams; or drawing the forces in on the provided figure. You donot need to include the force in the strap that connects the wheel axleto the ceiling – you may assume that it is large enough to keep the axleperfectly fixed.

b) When mass m1 falls a distance x1, by what distance x2 does mass m2

rise?

c) Find the tensions T1 and T2 in the ropes supporting m1 and m2.

d) After mass m1 falls a height H , what is its speed?

8.1. ROTATION 213

Problem 176.

problems/rotation-pr-atwoods-machine.tex

m1 m2

In the figure above Atwood’s machine is drawn – two masses m1 and m2 hangingover a massive pulley which you can model as a disk of mass M and radius R,connected by a massless unstretchable string. The string rolls on the pulleywithout slipping.

a) Draw three free body diagrams (isolated diagrams for each object showingjust the forces acting on that object) for the three masses in the figureabove.

b) Convert each free body diagram into a statement of Newton’s Second Law(linear or rotational) for that object.

c) Using the rolling constraint (that the pulley rolls without slipping as themasses move up or down) find the acceleration of the system and thetensions in the string on both sides of the pulley in terms of m1, m2, M ,g, and R.

d) Suppose mass m2 > m1 and the system is released from rest with themasses at equal heights. When mass m2 has descended a distance H , findthe velocity of each mass and the angular velocity of the pulley.


Problem 177.

problems/rotation-pr-bowling-ball-friction.tex

M0v vf

fωω = 00

R

d

A bowling ball of mass M and radius R is released horizontally moving at aspeed v0 so that it initially slides without rotating on the bowling lane floor. µk

is the coefficient of kinetic friction between the bowling ball and the lane floor.It slides for a time t and distance d before it rolls without slipping the rest ofthe way to the pins at speed vf .

a) Find t.

b) Find d.

c) Find vf .

8.1. ROTATION 215

Problem 178.

problems/rotation-pr-cable-spool-rolls-on-line-2.tex

R

F

R/2

A cable spool of mass M , radius R and moment of inertia I = 13MR2 is wrapped

around its OUTER disk with fishing line and set on a rough rope as shown. Thefishing line is then pulled with a force F to the right as shown so that it rollsdown the rope without slipping.

a) Which way does the spool roll (left or right)?

b) Find the magnitude of the acceleration of the spool.

c) Find the force the friction of the rope exerts on the spool.


Problem 179.

problems/rotation-pr-cable-spool-rolls-on-line.tex

Rr

F

A cable spool of mass M , radius R and moment of inertia I = βMR2 around anaxis through its center of mass is wrapped around its OUTER disk with fishingline and set on a rough rope as shown. The fishing line is then pulled with aforce F to the right as shown so that it rolls down the rope on the spool atradius r without slipping.

a) Which way does the spool roll (left or right)?

b) Find the magnitude of the acceleration of the spool.

c) Find the force the friction of the rope exerts on the spool.

d) Is there a value of the radius r relative to R for which friction exerts no

force on the spool? If so, what is it?

8.1. ROTATION 217

Problem 180.

problems/rotation-pr-disk-on-ice-example.tex

icy

rough

m,R

H


• Conservation of Mechanical Energy

• Rotational Kinetic Energy

• Rolling Constraint.


A disk of mass m and radius R rolls without slipping down a rough slope ofheight H onto an icy (frictionless) track at the bottom that leads up a secondicy/frictionless hill as shown.

a) How fast is the disk moving at the bottom of the first incline? How fastis it rotating (what is its angular velocity)?

b) Does the disk’s angular velocity change as it leaves the rough track andmoves onto the ice (in the middle of the flat stretch in between the hills)?

c) How far up the second hill (vertically, find H ′) does the disk go before itstops rising?


Problem 181.

problems/rotation-pr-disk-on-ice-solution.tex

icy

rough

m,R

H


• Conservation of Mechanical Energy

• Rotational Kinetic Energy

• Rolling Constraint.


A disk of mass m and radius R rolls without slipping down a rough slope ofheight H onto an icy (frictionless) track at the bottom that leads up a secondicy/frictionless hill as shown.

a) How fast is the disk moving at the bottom of the first incline? How fastis it rotating (what is its angular velocity)?

As the disk rolls down the incline without slipping, the velocity of itscenter and the angular velocity with which it rotates are related by therolling without slipping constraint:

v = ωR . (8.1)

Because it is not slipping, friction does no work, so that the total mechan-ical energy is conserved during the descent. Initial kinetic energy is zero,so Ei = Ui = mgH , where I am setting U = 0 at the bottom of the incline.With this choice, final potential energy vanishes and total energy in finalstate is kinetic. This, in turn, is a sum of a translational term repre-senting the motion of the center of mass and a rotational contributionrepresenting the motion as seen by an observer moving with the center ofmass. Thus:

8.1. ROTATION 219

Ef = Kf =1

2mv2 +

1

2Iω2 . (8.2)

Inserting the constraint as well as the value of the moment of inertia fora uniform disk I = 1

2mR2 we have

mgH =1

2mv2 +

1

2

mR2

2

( v

R

)2

=1

2mv2 (1 + 1/2) =

3

4mv2 , (8.3)

whence

v =

√

4

3gH . (8.4)

Note that this is less than the√

2gH we would find were the disk slidingdown a frictionless incline. This makes sense, because friction has beenacting to enforce the constraint, but has also slowed the disk. Alterna-tively, the expression for the kinetic energy above shows that some of thework done by gravity was converted into rotational kinetic energy, leavingless of it to be converted to translational kinetic energy.

Using the constraint we then have

ω =v

R=

√

4gH

3R2. (8.5)

b) Does the disk’s angular velocity change as it leaves the rough track andmoves onto the ice (in the middle of the flat stretch in between the hills)?

During the disk’s accelerating descent down the incline, friction acted toretard the acceleration and increase the angular acceleration, in order tomaintain the condition of no slipping, but it did no work because it actsat the one point on the wheel that is always stationary with respect tothe ground. Instead it served to redistribute the gravitational potentialenergy between translational and rotational kinetic energy.

Once the horizontal stretch is reached, the disk continues at the constant

translational and angular velocity given by the values we computed above.Since these satisfy the rolling constraint and no energy is entering thesystem, friction does not act on the disk as it rolls along the horizontalrough stretch.

This is an important fact! A perfectly round wheel (with frictionlessbearings) rolling without slipping on a level surface experiences no frictionand does not slow down. This is why we use wheels!

When the disk moves onto the ice the change in the coefficient of frictionthus produces no change in its motion, since friction was not applying anyforce on the rough surface anyway.


c) How far up the second hill (vertically, find H ′) does the disk go before itstops rising?

As it begins to climb the second incline, the disk’s velocity decreases askinetic energy is converted to potential energy. As its motion acquires avertical component gravity is doing negative work on the disk and thisforce slows the disk. On the other hand, with no friction the only forceson the disk, gravity and the normal force, exert no torque about thedisk’s center so its angular velocity remains constant at the value foundabove. The disk slows as it climbs but continues spinning at a constantrate. When it comes to a stop at the highest point it can reach, its totalmechanical energy is:

EH′ = mgH ′ +1

2Iω2 . (8.6)

where its rotational kinetic energy is unchanged.

This is equal to the total energy found above since during the climb allwork was done by gravity, whence we find

mgH ′ =1

2mv2 =

1

2m

(

4

3gH

)

, (8.7)

or

H ′ =2

3H . (8.8)

The disk does not recover its original height, though energy is conserved,because the energy converted to rotational kinetic energy cannot, withoutfriction, be converted to potential energy. If we throw sand on the ice asthe disk comes to a halt, the resulting friction will act to propel the diskfarther up the hill, slowing its rotation and recovering this stored energy.

8.1. ROTATION 221

Problem 182.

problems/rotation-pr-disk-rolling-on-slab-difficult.tex

FM

m,R

θ

MF


• Newton’s Second Law (linear and rotational)

• Rolling Constraint

• Static and Kinetic Friction


A disk of mass m is resting on a slab of mass M , which in turn is resting on africtionless table. The coefficients of static and kinetic friction between the diskand the slab are µs and µk, respectively. A small force ~F to the right is appliedto the slab as shown, then gradually increased.

a) When ~F is small, the slab will accelerate to the right and the disk willroll on the slab without slipping. Find the acceleration of the slab, theacceleration of the disk, and the angular acceleration of the disk as thishappens, in terms of m, M , R, and the magnitude of the force F .

b) Find the maximum force Fmax such that it rolls without slipping.

c) If F is greater than this, solve once again for the acceleration and angularacceleration of the disk and the acceleration of the slab.

Hint: The hardest single thing about this problem isn’t the physics (which isreally pretty straightfoward). It is visualizing the coordinates as the center of


mass of the disk moves with a different acceleration as the slab. I have drawntwo figures above to help you with this – the lower figure represents a possibleposition of the disk after the slab has moved some distance to the right andthe disk has rolled back (relative to the slab! It has moved forward relative tothe ground! Why?) without slipping. Note the dashed radius to help you seethe angle through which it has rolled and the various dashed lines to help yourelate the distance the slab has moved xs, the distance the center of the diskhas moved xd, and the angle through which it has rolled θ. Use this relation toconnect the acceleration of the slab to the acceleration and angular accelerationof the disk.

If you can do this one, good job!

8.1. ROTATION 223

Problem 183.

problems/rotation-pr-falling-mass-spins-disk.tex

m

M

R

H

A disk of mass M and radius R placed on a frictionless table can rotate freelyabout a fixed frictionless spindle as shown in the figure. An Acme (massless,unstretchable) string is tightly wound around the disk and then passes over asmall frictionless pulley, where it is attached to a hanging mass m. At timet = 0 the hanging mass and disk are released from rest.

a) Find the tension T in the string while the mass is falling and thedisk is rotating.

b) Find the speed vm of the mass m when it has fallen a height H from itsinitial position.


Problem 184.

problems/rotation-pr-flat-plane-two-blocks-massive-pulley.tex

m2

1m

M

R

A mass m1 is attached to a second mass m2 by an Acme (massless, unstretch-able) string. m1 sits on a frictionless table; m2 is hanging over the ends of atable, suspended by the taut string from pulley of mass M and radius R. Attime t = 0 both masses are released.

a) Draw the force/free body diagram for this problem.

b) Find the acceleration of the two masses.

c) Find the angular acceleration of the pulley.

8.1. ROTATION 225

Problem 185.

problems/rotation-pr-gyroscope-torque.tex

R

F

A child spins a gyroscope with moment of inertia I and a frictionless pivot bywrapping a (massless, unstretchable) string of length L around it at a radius Rand then pulling the string with a constant force F as shown. Find:

a) The angular acceleration of the gyroscope while the string is being pulled.

b) The angular speed of the gyroscope as the string comes free (assume thatthe force F is exerted through the entire distance L).


Problem 186.

problems/rotation-pr-rolling-spool-pulled-right.tex

R

R/2

F

A spool of mass M , radius R and moment of inertia I = 13MR2 is wrapped

around its spindle (radius R/2) with fishing line and set on a rough table asshown. The line is then pulled with a force F as shown so that it rolls withoutslipping.

a) Which way does the spool roll (left or right)? Put another way, does it rollup the string or unroll the string?

b) Find the magnitude of the acceleration of the spool and the force exerted bythe table on the spool.

8.1. ROTATION 227

Problem 187.

problems/rotation-pr-rolling-wheel-static-friction-numbers.tex

F

M

R

A force ~F = 48 N to the right is applied to the frictionless axle of a wheel madeof a uniform disk with mass M = 4 kg and radius R = 10 cm. It rolls withoutslipping on a rough table. Find:

a) the net acceleration of the wheel.

b) The minimum coefficient of static friction µs such that the wheel does notslip for this force.


Problem 188.

problems/rotation-pr-rolling-wheel-static-friction.tex

F

M

R

A force of magnitude F (to the left) is applied to the frictionless axle of a wheelmade of a uniform disk with mass M and radius R. It rolls without slippingon a rough table (with a coefficient of static friction given by µs). Find:

a) The magnitude of the acceleration of the wheel.

b) The magnitude of the force exerted by static friction. Indicate its di-rection on the figure above.

c) The minimum coefficient of static friction µs such that the wheel does notslip for this force.

8.1. ROTATION 229

Problem 189.

problems/rotation-pr-sliding-rolling-bowling-ball.tex

A uniform bowling ball of radius R, mass M , and moment of inertia I about itscenter of mass is initially launched so that it is sliding with speed v0 withoutrolling on an alley with a coefficient of friction µk.

a) Analyze the forces acting on the bowling ball to find the acceleration ofthe center of mass and angular acceleration of the bowling ball about itsCM;

b) Find the CM velocity as a function of time (t) and angular velocity of theball as a function of time (t).

c) Find the CM velocity of the bowling ball when it starts rolling withoutslipping.


Problem 190.

problems/rotation-pr-two-spools-one-mass.tex

M M

M

T T

R R

In the figure above, a mass M is connected to two independent massive spoolsof radius R, also of mass M (each), wrapped with massless unstretchable string.You may consider the spools to be disks as far as their moment of inertia isconcerned. At t = 0, the mass M and spools are released from rest and themass M falls. Find:

a) The magnitude of the acceleration a of the mass M .

b) The tension T in either string (they are the same from symmetry).

c) When the mass M has fallen a distance H , what is its speed?

8.1. ROTATION 231

Problem 191.

problems/rotation-pr-unrolling-a-falling-spool-algebraic.tex

M

T

R

A spool of fishing line is tied to a beam and released from rest in the positionshown at time t = 0. The spool has a mass M , a radius of R, and a momentof inertial I = βMR2. The line itself has negligible mass per unit length. Oncereleased, the disk falls as the taut line unrolls.

a) What is the tension in the line as the disk falls (unrolling the line)?

b) After the disk has fallen a height H , what is its angular velocity ω?


Problem 192.

problems/rotation-pr-unrolling-a-falling-spool-numeric.tex

RM

A spool of fishing line is tied to a beam and released from rest in the positionshown at time t = 0. The spool is a disk and has a mass of 50 grams and a radiusof 5 cm. The line itself has negligible mass per unit length. Once released, thedisk falls as the taut line unrolls.

a) What is the tension in the line as the disk falls (unrolling the line)?

b) After the disk has fallen 2m, what is its speed?

8.1. ROTATION 233

Problem 193.

problems/rotation-pr-unrolling-a-falling-spool-reversed.tex

T

R

M

A spool of fishing line is tied to a beam and released from rest in the positionshown at time t = 0. The spool has a mass M , a radius of R, and a momentof inertial I = βMR2. The line itself has negligible mass per unit length. Oncereleased, the spool falls as the taut line unrolls.

a) What is the tension in the line as the spool falls (unrolling the line)?

b) What is the magnitude of the angular acceleration of the spool α aboutits center of mass as it falls?

c) After the spool has fallen a height H , what is the direction of its angularvelocity, ~ω? Indicate this direction with a labelled arrow symbol on asuitable diagram.


Problem 194.

problems/rotation-pr-unrolling-a-falling-yo-yo.tex

M

R

T

R/2

A yo-yo is tied to a beam and released from rest in the position shown attime t = 0. The yo-yo has a mass M , a radius of R, and a moment of inertiaI = βMR2. The unstretchable line itself has negligible mass per unit length andis wrapped around an inner spindle with radius R/2 as shown. Once released,the yo-yo falls as the taut line unrolls.

a) What is the angular acceleration ~α of the yo-yo as it falls (unrolling theline)? Note that this is a vector quantity, so please indicate its directionin your answer and/or on the figure.

b) What is the tension T in the line as the yo-yo falls (unrolling the line)?

c) After the yo-yo has fallen a height H , what is its angular velocity ω?

8.1. ROTATION 235

Problem 195.

problems/rotation-pr-unrolling-disk-and-block.tex

Rr

M

M

Challenge Problem (difficult!): In the figure above, a spool with momentof inertia βMR2 is hanging from a rod by a (massless, unstretchable) stringthat is wrapped around it at a radius R, while a block of equal mass M is hungon a second string that is wrapped around it at a radius r as shown. Find themagnitude of the acceleration of the the central pulley.


Problem 196.

problems/rotation-pr-unrolling-spool-down-inclined-plane-1.tex

M

R

θ

ω = 0

H

m

v = 0

At t = 0

In the figure above, a pulley at rest of mass M and radius R with frictionlessbearings and moment of inertia I = βMR2 is fixed at the top of a fixed, friction-less inclined plane that makes an angle θ with respect to the horizontal. Thepulley is wrapped with many turns of (approximately massless and unstretch-able) fishing line. The line is also attached to a block of mass m. At time t = 0the block and pulley are released from rest , v = 0 (block) and ω = 0 (pulley).

• Find the magnitude of the acceleration a of the block as it slides downthe incline.

• Find the tension T in the string as it slides.

• Find the speed v with which the block reaches the bottom of the incline.

8.1. ROTATION 237

Problem 197.

problems/rotation-pr-unrolling-spool-down-inclined-plane-2.tex

θ

H

R

Mm

µk

In the figure above, a pulley of mass M and radius R with frictionless bearingsand moment of inertia I = βMR2 is fixed at the top of a rough inclined planethat makes an angle θ with respect to the horizontal that is large enough thatthe block will definitely overcome static friction and slide. The coefficient ofkinetic friction between the block and the plane is µk. The pulley is wrappedwith many turns of (approximately massless and unstretchable) fishing line.The line is also attached to a block of mass m. At time t = 0 the block andpulley are released from rest .

a) Draw a force diagram for both the block and the pulley separately.You do not have to represent the forces acting at the pivot of the pulleythat keep it stationary, only the one(s) relevant to the solution of theproblem. Represent all the forces on the block.

b) Find both the magnitude of the acceleration a of the block and the tensionT in the string as the block slides down the incline in terms of the givens.

c) Find the kinetic energy of the block when it reaches the bottom of theincline.

d) Find the kinetic energy of the pulley when the block reaches the bottomof the incline.


Problem 198.

problems/rotation-pr-unrolling-spool-down-inclined-plane-friction.tex

M

R

θ

ω = 0

H

m

v = 0

At t = 0

A pulley with frictionless bearings and moment of inertia I = βMR2 is atthe top of a fixed inclined plane that makes an angle θ with respect to thehorizontal. The pulley is wrapped with many turns of (approximately masslessand unstretchable) fishing line that is attached to a block of mass m restingon the incline a height H above the bottom. The coefficient of kinetic frictionbetween the block and the incline is µk. At time t = 0 the block and pulleyare released from rest at an angle θ that is large enough that the block willdefinitely overcome static friction and begin to slide.

a) On the figure above or in a free body diagram to the side, draw in andlabel all of the forces acting on the block only.

b) Find the magnitude of the acceleration a of the block as it slides downthe incline.

c) Find the speed v with which the block reaches the bottom of the incline.

8.1. ROTATION 239

Problem 199.

problems/rotation-pr-walking-the-spool.tex

F

R

r

mass MA FB C

F

In the figure above, a spool of mass m is wrapped with string around the innerspool. The spool is placed on a rough surface and the string is pulled with forceF in the three directions shown. The spool, if it rolls at all, rolls withoutslipping. (Note that if pulled too hard, the spool can both slip and/or roll.)

Find the acceleration and frictional force vectors (magnitude and direction)for all three figures. Use Icm = βMR2.

Note Well: You can use either the center of mass or the point of contactwith the ground (with the parallel axis theorem) as a pivot, the latter beingslightly easier.


8.2 Moment of Inertia


Problem 200.

problems/moment-of-inertia-ra-parallel-axes.tex

cm

A B

C

D

A two-dimensional cardboard cut-out of an elephant is drawn above. Small holesare drilled through it at the points A, B, C and D indicated. Hole C is at thecenter of mass of the figure. Rank the moment of inertia of the elephant aboutaxes through each of the holes (with equality permitted) so that a possible (butunlikely) answer is IA < IB = IC < ID.

8.2. MOMENT OF INERTIA 241

Problem 201.

problems/moment-of-inertia-ra-point-masses-massless-rods-2.tex

r = 1m

2m

= 1 kg

3m

A

B

D

1m rotation axes

C

In the figure above, all of the masses m are identical and are connected by rigidmassless rods as drawn. Rank the moments of inertia of the four objectsabout the rotation axes drawn as dashed lines. Equality is permitted, soa possible answer might be A > C = D > B.


Problem 202.

problems/moment-of-inertia-ra-point-masses-massless-rods.tex

A B

C

D

m

rotation axes1

r = 1

2

3

In the figure above, all of the masses m are identical and are connected by rigidmassless rods as drawn. Rank the moments of inertia of the four objects aboutthe rotation axes drawn as dashed lines. Equality is permitted, so a possibleanswer might be A > C = D > B.


8.2.2 Short Answer

Problem 203.

problems/moment-of-inertia-sa-3-masses.tex

3mm

pivot

a

a

y

x2m

(z−axis)

In the figure above, massless rigid rods connect three masses to a pivot at theorigin so that they can freely rotate around the z-axis (perpendicular to thepage). The masses are fixed so that they are at three corners of a square of sidea, with the pivot at the fourth corner as shown. Find the moment of inertiaabout the z-axis of this arrangement in terms of m and a.


Problem 204.

problems/moment-of-inertia-sa-4-masses-2.tex

2 kg

−5m

5m

x

y

2 kg 2 kg

2 kg

−5m 5m4 kg

In the figure above, massless rigid rods connect five masses to form a rigid body.The rigid body can be rotated about any axis perpendicular to the plane of thefigure. Find a point with coordinates (x0, y0) on the provided coordinate frame(units of meters) so that the moment of inertia of the system is smallest if theaxis goes through this point. Then, enter the moment of inertia of the systemabout this axis.

x0 = meters

y0 = meters

Imin = kg-meter2


Problem 205.

problems/moment-of-inertia-sa-4-masses.tex

In the figure below, four 2 kilogram masses are held at the corners of a rigidsquare by massless rods as shown. The center of mass of the system is locatedat the origin of the provided x − y coordinate frame. The z-axis points out ofthe page.

2 kg

−5m

5m

x

y

2 kg 2 kg

2 kg

2nd pivotcm

−5m 5m

Find the moment of inertia of this system around the z-axis through the centerof mass:

Icm =

Now find the moment of inertia of this system around an axis parallel to thez-axis but passing through a new pivot point at (−5, 0, 0) meters.

Inew =


Problem 206.

problems/moment-of-inertia-sa-disk-axis-in-plane.tex

RM

axis

In the figure above, a disk of mass M and radius R is rotated around an axisthrough the middle in the plane of the ring as shown. Find the moment ofinertia of the disk about this axis using the perpendicular axis theorem.


8.2.3 Long Problems

Problem 207.

problems/moment-of-inertia-pr-disk-pivoted-at-rim.tex

RM

pivot

In the figure above, a disk of mass M and radius R is pivoted about a point onthe rim as shown. What is the moment of inertia of the disk about this pivot?


Problem 208.

problems/moment-of-inertia-pr-disk-with-holes.tex

R

R/2

b

a

ρ

You are employed by a company that makes cogs, pulleys, and other widgets forlawnmower engines. They have designed a new pulley that is basically an annu-lar disk of thickness t, outer radius R, and inner radius a, with approximatelyuniform density otherwise, as shown. To save on material costs (and to be ableto deliver more torque to the real payload, instead of the pulley itself) they haveremoved all the material in four large circular holes of radius b through the solidpart of the disk, centered on a circle of radius R/2 as shown. Your job is tocompute the new moment of inertia as a function of ρ, t, R and a, b < R/2.

Hints: Note that you SHOULDN’T have to actually do any integrals in thisproblem if you remember that the moment of inertia of a disk is 1

2MR2. Youare also welcome to introduce quantities like M = ρπR2t, ma = ρπa2t and mb =ρπb2t into the problem if it would make the final answer simpler. Explain/showyour reasoning regardless.


Problem 209.

problems/moment-of-inertia-pr-moments-of-a-leg.tex

x 0

L

x

pivotdx


• Finding the Center of Mass using Integration

• Finding the Moment of Inertia using Integration


A simple model for the one-dimensional mass distribution of a human leg oflength L and mass M is:

λ(x) = C · (L + x0 − x)

Note that this quantity is maximum at x = 0, varies linearly with x, and vanishessmoothly at x = L + x0. That means that it doesn’t reach λ = 0 when x = L,just as the mass per unit length of your leg doesn’t reach zero at your ankles.

a) Find the constant C in terms of M , L, and x0 by evaluating:

M =

∫ L

0

λ(x) dx

and solving for C.

b) Find the center of mass of the leg (as a distance down the leg from thehip/pivot at the origin). You may leave your answer in terms of C (nowthat you know it) or you can express it in terms of L and x0 only as youprefer.

c) Find the moment of inertia of the leg about the hip/pivot at the origin.Again, you may leave it in terms of C if you wish or express it in termsof M , L and x0. Do your answers all have the right units?

d) How might one improve the estimate of the moment of inertia to take intoaccount the foot (as a lump of “extra mass” mf out there at x = L thatdoesn’t quite fit our linear model)?


This is, as you can see, something that an orthopedic specialist might well needto actually do with a much better model in order to e.g. outfit a patient with anartificial hip. True, they might use a computer to do the actual computationsrequired, but is it plausible that they could possibly do what they need to dowithout knowing the physics involved in some detail?


Problem 210.

problems/moment-of-inertia-pr-ring-axis-in-plane.tex

RM

axis

In the figure above, a ring of mass M and radius R is rotated around an axisthrough the middle in the plane of the ring as shown.

a) Find the moment of inertia of the ring about this axis through directintegration.

b) Find the moment of inertia of the ring about this axis using the perpen-dicular axis theorem. Which is easier?


Problem 211.

problems/moment-of-inertia-pr-ring-pivoted-at-rim.tex

RM

pivot

In the figure above, a ring of mass M and radius R is pivoted about a point onthe rim as shown. What is the moment of inertia of the ring about this pivot?

Chapter 9

Vector Torque and AngularMomentum

Well, if we can rotate around an axis in the x, y, or z direction, we can rotatearound an axis in any direction. So I guess torque has to be a vector quantity!Since it already has the magnitude of the cross product ~r× ~F , we might as welldefine that to be the vector torque.

That, in turn, is related to the new quantity ~L = ~r×~p, the angular momen-tum (also a vector quantity) and suddenly we can do rotational collisionsthat conserve angular momentum where before we did linear collisions thatconserved regular vector momentum. Or we can even do collisions that conserveboth!

The most startling thing about vector torque, however, is when we observe aspinning object precess under the application of a vector torque. Our defini-tions above work perfectly to describe an insanely complicated motion(if you think about it) in simple terms. Ones we can compute. Ones that I, atleast, require students to be able to solve for and understand.

Too bad Obi-wan didn’t say Use the Torque, Luke!”...

253

254 CHAPTER 9. VECTOR TORQUE AND ANGULAR MOMENTUM

9.1 Angular Momentum


Problem 212.

problems/angular-momentum-mc-collapsing-star.tex

before

after

ω

ω f

i

When a star rotating with an angular speed ωi (eventually) exhausts its fuel,escaping light energy can no longer oppose gravity throughout the star’s volumeand it suddenly shrinks, with most of its outer mass falling in towards thecenter all at the same time.

As this happens, does the magnitude of the angular speed of rotation ωf :

a) increase

b) decrease

c) remain about the same

Why (state the principle used to answer the question)?

9.1. ANGULAR MOMENTUM 255

Problem 213.

problems/angular-momentum-mc-forming-star.tex

ω f

after

ω i

before

2RR

Gravity gradually assembles a star by pulling a cloud of rotating gas togetherinto a rotating ball that then gradually shrinks. The figure above represents astar at two different stages in its formation, the first where a gas of total massM has formed a ball of radius 2R rotating at angular speed ωi, the second wherethe ball has collapsed to a radius R (compressing the nuclear fuel inside closerto the point of fusion and ignition), rotating at a possibly new angular speedωf .

Assuming that the mass is uniformly distributed in both cases, what is the bestestimate for ωf in terms of ωi?

a) ωf = ωi

b) ωf = 2ωi

c) ωf = 4ωi

d) ωf = ωi/2

e) ωf = ωi/4


Problem 214.

problems/angular-momentum-mc-rotating-rod-sliding-beads.tex

L/2

ω

−L/2

i

In the figure above, a massless rod of length L is rotating around a frictionlesspivot through its center at angular speed ωi. Two beads, each with mass m, arestuck a distance L/4 from the center. The rotating system initially has a totalkinetic energy Ki (which you could actually calculate if you needed to). At acertain time, the beads are released and slide smoothly to the ends of the rodwhere again, they stick. Which statement about the final angular speed androtational kinetic energy of the rotating system is true:

a) ωf = ωi/2 and Kf = Ki.

b) ωf = ωi/4 and Kf = Ki/4.

c) ωf = ωi/4 and Kf = Ki/2.

d) ωf = ωi/2 and Kf = Ki/4.

e) ωf = ωi/2 and Kf = Ki/2.


9.1.2 Short Answer

Problem 215.

problems/angular-momentum-sa-conserved-quantities.tex

For each of the collisions described below, say whether the total mechanicalenergy, total momentum, and total angular momentum of the system consistingof the two colliding objects are conserved or not. Indicate your answer by writing“C” (for “is definitely conserved”) or “N” (for “not necessarily conserved”)in each box. You may write a brief word of explanation if you think there isany ambiguity in the answer.

Total Linear AngularEnergy Momentum Momentum

A hard ball (point particle) bounces off of a rigidwall that cannot move, returning at the samespeed it had before the collision.

A piece of space junk strikes the orbiting spaceshuttle and sticks to it.


Problem 216.

problems/angular-momentum-sa-rotating-rod-sliding-beads.tex

L/2

ω

−L/2

i

In the figure above, a massless rod of length L is rotating around a frictionlesspivot through its center at angular speed ωi. Two beads, each with mass m, arestuck a distance L/4 from the center. The rotating system initially has a totalkinetic energy Ki (which you could actually calculate if you needed to). At acertain time, the beads are released and slide smoothly to the ends of the rodwhere again, they stick.

A) What quantities of the system (rod plus two beads) are conserved by thisprocess? (Place a Y or N in the provided answer boxes.)

Total Kinetic Energy

Total Linear Momentum

Total Angular Momentum

B) Determine the ratio of the following quantities:

If

Ii=

ωf

ωi=

Kf

Ki=


9.1.3 Long Problems

Problem 217.

problems/angular-momentum-pr-circular-orbit-on-table.tex

rm

F

v


• Torque Due to Radial Forces

• Angular Momentum Conservation

• Centripetal Acceleration

• Work and Kinetic Energy


A particle of mass M is tied to a string that passes through a hole in a frictionlesstable and held. The mass is given a push so that it moves in a circle of radiusr at speed v. We will now analyze the physics of its motion in two stages.

a) What is the torque exerted on the particle by the string? Will angularmomentum be conserved if the string pulls the particle into “orbits” withdifferent radii?

b) What is the magnitude of the angular momentum L of the particle in thedirection of the axis of rotation (as a function of m, r and v)?

c) Show that the magnitude of the force (the tension in the string) that mustbe exerted to keep the particle moving in a circle is:

F =L2

mr3

This is a general result for a particle moving in a circle and in no waydepends on the fact that the force is being exerted by a string in particular.

d) Show that the kinetic energy of the particle in terms of its angular mo-mentum is:

K =L2

2mr2


Now, suppose that the radius of the orbit and initial speed are ri and vi, re-spectively. From under the table, the string is slowly pulled down (so that thepuck is always moving in an approximately circular trajectory and the tensionin the string remains radial) to where the particle is moving in a circle of radiusr2.

e) Find its velocity v2 using angular momentum conservation. This shouldbe very easy.

f) Compute the work done by the force from part c) above and identify theanswer as the work-kinetic energy theorem. Use this to to find the velocityv2. You should get the same answer!

Note that the last two results are pretty amazing – they show that our torqueand angular momentum theory so far is remarkably consistent since two very

different approaches give the same answer. Solving this problem now will makeit easy later to understand the angular momentum barrier, the angular kineticenergy term that appears in the radial part of conservation of mechanical energyin problems involving a central force (such as gravitation and Coulomb’s Law).This in turn will make it easy for us to understand certain properties of orbitsfrom their potential energy curves.


Problem 218.

problems/angular-momentum-pr-collapse-of-sun.tex

R

R

i

f

before

after

m

m

The sun reaches the end of its life and gravitationally collapses quite suddenly,forming a white dwarf. Before it collapses, it has a mass m, a radius Ri, anda period of rotation Ti. After it collapses, its radius is Rf ≪ Ri and we willassume that its mass is unchanged. We will also assume that before and after themoment of inertia of the sun is given by I = βmR2 where R is the appropriateradius.

a) What is its final period of rotation Tf after the collapse?

b) Evaluate the escape velocity from the surface of the sun before and afterits collapse.

For 2 points of extra credit, evaluate the numbers associated with these expres-sions given β = 0.25, m = 2 × 1030 kg, Ri = 5 × 105 km, Rf = 100 km, andTi = 108, 000 seconds. These numbers are actually quite interesting in cosmol-ogy, as the escape velocity from the surface of the white dwarf approaches thespeed of light...


Problem 219.

problems/angular-momentum-pr-disk-collides-with-pivoted-rod.tex

L

L/4

ω

v = 0f

fω = 0

0

0

M

mv

initial final

A steel rod of mass M and length L with a frictionless pivot in the centerand moment of inertia 1

12ML2 sits on a frictionless table at rest. The pivot isattached to the table. A steel disk of mass m approaches with velocity v0 fromthe left and strikes the rod a distance L/4 from the lower end as shown. Thiselastic collision instantly brings the disk to rest and causes the rod to rotatewith angular velocity ωf .

a) What quantities are conserved in this collision?

b) Find the angular velocity ωf of the rod about the pivot after the collision.

c) Find the ratio m/M such that the collision occurs elastically, as de-scribed.


Problem 220.

problems/angular-momentum-pr-marble-and-rod.tex

L/2

Lm

m

v0

In the figure above, a marble with mass m travelling to the right at speed v0

collides with a rigid rod of length L pivoted about one end, also of mass m, .The marble strikes the rod L/2 down from the pivot and comes precisely to restin the collision. Ignore gravity, drag forces, and any friction in the pivot.

a) What is the rotational velocity of the rod after the collision?

b) What is the change in linear momentum during this collision?

c) What is the energy gained or lost in this collision?


Problem 221.

problems/angular-momentum-pr-putty-sticks-to-pivoted-rod-gravity.tex

vm

d

L

M

A rod of mass M and length L is hanging vertically from a frictionless pivot(where gravity is “down”). A blob of putty of mass m approaches with velocityv from the left and strikes the rod a distance d from its center of mass as shown,sticking to the rod.

a) Find the angular velocity ωf of the system about the pivot (at the top ofthe rod) after the collision.

b) Find the distance xcm from the pivot of the center of mass of the rod-puttysystem immediately after the collision.

c) After the collision, the rod swings up to a maximum angle θmax and thencomes momentarily to rest. Find θmax.

All answers should be in terms of M , m, L, v, g and d as needed. The momentof inertia of a rod pivoted about one end is I = 1

3ML2, in case you need it.


Problem 222.

problems/angular-momentum-pr-putty-sticks-to-pivoted-rod.tex

vm

d

L

M




• Inelastic Collisions

• Impulse

so please review them before you begin. You may also find it useful to read:Wikipedia: http://www.wikipedia.org/wiki/Center of Percussion.

A rod of mass M and length L rests on a frictionless table and is pivoted ona frictionless nail at one end as shown. A blob of putty of mass m approacheswith velocity v from the left and strikes the rod a distance d from the end asshown, sticking to the rod.

• Find the angular velocity ω of the system about the nail after the collision.

• Is the linear momentum of the rod/blob system conserved in this collisionfor a general value of d? If not, why not?

• Is there a value of d for which it is conserved? If there were such a value,it would be called the center of percussion for the rod for this sort ofcollision.

All answers should be in terms of M , m, L, v and d as needed. Note well thatyou should clearly indicate what physical principles you are using to solve thisproblem at the beginning of the work.

http://www.wikipedia.org/wiki/Center of Percussion


Problem 223.

problems/angular-momentum-pr-putty-sticks-to-unpivoted-rod.tex

mv

M

L

d




• Inelastic Collisions

• Impulse


A rod of mass M and length L rests on a frictionless table. A blob of putty ofmass m approaches with velocity v from the left and strikes the rod a distanced from the end as shown, sticking to the rod.

• Find the angular velocity ω of the system about the center of mass of the

system after the collision. Note that the rod and putty will not be rotatingabout the center of mass of the rod!

• Is the linear momentum of the rod/blob system conserved in this collisionfor a general value of d? If not, why not?

All answers should be in terms of M , m, L, v and d as needed. Note well thatyou should clearly indicate what physical principles you are using to solve thisproblem at the beginning of the work.


Problem 224.

problems/angular-momentum-pr-spinning-cups-catch-balls.tex

ω0

m

m

M

L

In the figure above, a bar of length L with two cups at the ends is freely rotating(in space – ignore gravity and friction or drag forces) about its center of masswith angular velocity ω0. The bar and cups together have a mass M and amoment of intertia of I = βML2. When the bar reaches the vertical position,the cups catch two small balls of mass m that are at rest, which stick in thecups. The balls have a negligible moment of inertia about their own center ofmass – you may think of them as particles.

a) What is the velocity of the center of mass of the system after the collision?

b) What is the angular velocity of the bar after it has caught the two ballsin its cups? Is kinetic energy gained or lost in this process?


Problem 225.

problems/angular-momentum-pr-swinging-rod-strikes-putty.tex

L

m

m

A uniform rod of mass m and length L swings about a frictionless peg throughits end. The rod is held horizontally and released from rest as shown in thefigure. At the bottom of its swing the rod strikes a ball of putty of mass mthat sits at rest on a frictionless table. In answering the questions take themagnitude of acceleration due to gravity to be g and assume that gravity actsdownward (in the usual way). The questions below should be answered in termsof the given quantities.

a) What is the angular speed ωi of the rod just before it hits the putty?

b) If the putty sticks to the rod, what is the angular speed ωf of the rod-putty system immediately after the collision?

c) What is ∆E, the mechanical energy change of the system in this collision(be sure to specify its sign).


Problem 226.

problems/angular-momentum-pr-two-circular-plates-collide.tex

R

M

x

yz

0ω

A disk of mass M and radius R sits at rest on a turntable that permits it torotate freely. A second identical disk, this one rotating around their mutual axisat an angular speed ω0, is dropped gently onto it so that (after sliding for aninstant) they rotate together.

a) Find the final angular speed ωf of the two disks moving together after

the collision.

b) What fraction of the original kinetic energy of the system is lost in thisrotational collision.


9.2 Vector Torque


Problem 227.

problems/torque-vector-mc-rotational-collision-two-disks.tex

R

M

x

yz

0ω

Two identical disks with mass M and radius R have a common axis and fric-tionless bearing. Initially, one disk is spinning with some angular velocity ω0

and the other is rest. The two disks are brought together quickly so that theystick and rotate as one without the application of any external torque. Circlethe true statement below:

a) The total kinetic energy and the total angular momentum are unchanged.

b) The total kinetic energy and total angular momentum are both reducedto half their original values.

c) The total kinetic energy is unchanged, but the total angular momentumis reduced to half of its original value.

d) The total angular momentum is unchanged, but the total kinetic energyis reduced to half of its original value.

e) We cannot tell what happens to the angular momentum and kinetic energyfrom the information given.

9.2. VECTOR TORQUE 271

9.2.2 Short Answer

Problem 228.

problems/torque-vector-sa-bug-on-rotating-disk.tex

A disk of mass M and radius R is rotating about its axis with initial angularvelocity ω0. A rhinoceros beetle with mass m is standing on its outer rim as itdoes so. The beetle decides to walk in to the very center of the disk and standon the axis as it feels less pseudoforce there and it is easier to hold on. What isthe angular velocity of the disk when it gets there?

(Ignore friction and drag forces).


Problem 229.

problems/torque-vector-sa-direction-of-precession-1.tex

b)a) d)c)

In the figure above four symmetric gyroscopes are portrayed. Each gyroscopeis spinning very rapidly in the direction shown, and is suspended/pivoted fromone end as shown at the big arrow (gravity points down). For each figure a-dindicate whether the gyroscope will precess in or out of the page at the other

(non-pivoted, free) end at the instant shown.


Problem 230.

problems/torque-vector-sa-direction-of-precession-2.tex

b)a) d)c)

In the figure above four symmetric gyroscopes are portrayed. Each gyroscopeis spinning very rapidly in the direction shown, and is suspended/pivoted fromone end as shown at the big arrow (gravity points down). For each figure a-dindicate whether the gyroscope will precess in or out of the page at the other

(non-pivoted, free) end at the instant shown.


Problem 231.

problems/torque-vector-sa-direction-of-precession.tex

In the figure above four symmetric gyroscopes are portrayed. Each gyroscope isspinning very rapidly in the direction shown, and is suspended from one end asshown (at the big arrow). For each figure indicate whether the gyroscope willprecess in or out of the page at the other (free) end at the instant shown.


Problem 232.

problems/torque-vector-sa-evaluate-the-torque-1.tex

Fy

xF

x

y

In the figure above, a force~F = 2x− 1y

Newtons is applied to a disk at the point

~r = 2x + 2y

as shown. (That is, Fx = 2 N, Fy = −1 N, x = 2 m, y = 2 m). Find the total

torque about a pivot at the origin.

Don’t forget that torque is a vector, so either give the answer in cartesiancoordinates or otherwise specify its direction!


Problem 233.

problems/torque-vector-sa-evaluate-the-torque-2.tex

x

y

Fx

Fy

r F

In the figure above, a force~F = 2x + 1y

Newtons is applied to a disk at the point

~r = 2x − 2y

as shown. (That is, Fx = 2 N, Fy = 1 N, x = 2 m, y = −2 m). Find thetotal torque about a pivot at the origin. Don’t forget that torque is a vector, sospecify its direction as well as its magnitude (or give the answer as a cartesianvector)! Show your work!


Problem 234.

problems/torque-vector-sa-precession-of-top-1.tex

ω

Clearly show the direction of ~L on the spinning top in the figure above, andclearly indicate the direction that the top will precess (in or out of the page).


Problem 235.

problems/torque-vector-sa-precession-of-top-2.tex

L

ω

Clearly show the direction that the spinning top will precess on the figure above,given the direction of its angular momentum as indicated.


9.2.3 Long Problems

Problem 236.

problems/torque-vector-pr-crane-boom.tex

pivot

boom

y

xz (out)

30 45

M

T

L

sin(30) = cos(60) =1

2

cos(30) = sin(60) =

√3

2

sin(45) = cos(45) =

√2

2

A crane with a “massless” boom (the long support between the body and theload) of length L holds a mass M suspended as shown. Note that the wire withthe tension T is fixed to the top of the boom, not run over a pulley to the massM .

a) Find the torque (magnitude and direction) exerted by the tension in thewire on the boom, relative to a pivot at the base of the boom.

b) Find the torque (magnitude and direction) exerted by the hanging mass,relative to a pivot at the base of the boom.


Problem 237.

problems/torque-vector-pr-precessing-bicycle-wheel.tex

Ω

d

R

M

O

A bicycle wheel (basically a ring) of mass M and radius R has massless spokesand a massless axle of length d. The other end O of the axle rests on a conicalsupport as shown. The axle is held in a horizontal position and the wheel isspun with a large angular velocity ~Ω that points towards O, and then releasedso that the wheel precesses about O.

(Note: To specify the direction of vectors you may use up, down, towards O,away from O, into the page, out of the page as shown.)

a) What is the angular momentum ~L of the wheel about its center of mass?

b) What is the angular frequency of precession ~ωp of the wheel?

c) What is the kinetic energy K of the wheel in the frame of O (i.e., the labframe)?


Problem 238.

problems/torque-vector-pr-precession-of-equinoxes.tex

ecliptic north

rotationalnorth

ω

θ

τav

The Earth revolves on its axis. Its north (right handed) axis is significantlytipped relative to the “ecliptic” pole of the Earth’s revolution around the Sun.It is currently aligned with Polaris, the pole star, but because the Sun exertsa small torque on it due to tides acting on its slightly oblate spheroidal shape,it also precesses around the ecliptic north once every (approximately) 26,000years!

a) Assuming that the average torque on the earth over the course of anygiven year remains perpendicular to its angular momentum in the direc-tion/handedness shown, derive an algebraic expression for the angularfrequency of precession in terms of the magnitude of the torque. You mayuse I as the moment of inertia for the earth about its rotational axis asthat quantity is given below.

b) Given the data that the moment of inertia of the Earth about its axis ofrotation is roughly 8 × 1037 kg-m2, that its axis is tipped at roughly 20degrees relative to the ecliptic and that its period of revolution about itsown axis is one day, estimate the approximate magnitude of the averagetorque exerted by the Sun on the Earth over the course of a year.

(You may find it useful to know that 1 day = 86400 seconds, and 1 year =3.15 × 107 seconds – you can remember the latter as approximately π × 107

seconds.)


Problem 239.

problems/torque-vector-pr-precession-of-spherical-top.tex

M.R

θ

D

ω

A top is made from a ball of radius R and mass M with a very thin, light nail(r ≪ R and m ≪ M) for a spindle so that the center of tha ball is a distanceD from the tip. The top is spun with a large angular velocity ω, and has amoment of inertia I = 2

5MR2.

a) What is the angular momentum of the spinning ball? Indicate its (vector)direction with an arrow on the figure.

b) When the top is spinning at a small angle θ with the vertical (as shown)what is the angular frequency ωp of the top’s precession?

c) Does the top precess into or out of the page at the instant shown?


Problem 240.

problems/torque-vector-pr-precession-of-top-3-parts.tex

pivot

x

y

z

M.R

ω

θ

D

A top is made of a uniform disk of radius R and mass M with a very thin, light(assume massless) nail for a spindle so that the center of the disk is a distanceD from the tip. The top is spun with a large angular velocity ω with the nailvertically above the y-axis as shown above.

a) Find the vector torque ~τ exerted about the pivot at the instant shown inthe figure. You may express the vector however you wish (e.g. magnitudeand direction, cartesian components).

b) What is the axis of precession?

c) Derive the precession frequency ωp. Any of the derivations used in classor discussed in the textbook are acceptable.

Express all answers in terms of M, R, g, D, and θ as needed.


Problem 241.

problems/torque-vector-pr-precession-of-top.tex

M.R

ω

θ

ωp

D


• Vector Torque

• Vector Angular Momentum

• Geometry of Precession


A top is made of a disk of radius R and mass M with a very thin, light nail(r ≪ R and m ≪ M) for a spindle so that the disk is a distance D from the tip.The top is spun with a large angular velocity ω. When the top is spinning at asmall angle θ with the vertical (as shown) what is the angular frequency ωp ofthe top’s precession?


Problem 242.

problems/torque-vector-pr-rotating-bar-elastic-collision-balls.tex

v

ω0

f

vf

L/2

M

m

m

at rest after collision

In the figure above, a solid rod of length L and mass M is spinning aroundits center of mass. It then simultaneously strikes two hard balls of mass m adistance L/2 from the center of rotation as shown, causing them to elasticallyrecoil to the right. After the collision the rod is at rest.

a) Is momentum conserved in this collision?

b) Find the ratio m/M such that the problem description given above is true.

Chapter 10

Statics

Ah, time for a rest. We’ve spent a lot of energy (now that we know what itis) learning how to solve problems where lots of stuff moves and acceleratesand does all kinds of dynamical things. Let’s think about systems where theinteresting thing is that nothing happens.

We actually really need a lot of these systems, and yeah, they involve a fair bitof physics. Engineers do better if they build bridges and buildings that don’t

fall down. Physicians like for their patients not to fall over. Human beingslike to hang pictures, see-saw with their kids, arm wrestle, put things on tables,carry around glasses full of beer, balance things on their heads, build houses ofcards and so much more where they idea is that these things should not move,or tip over, or fall down, or snap a suspending wire.

In order to sit still, an object has to start out sitting still and not accelerate(linearly or angularly). So a necessary condition for static equilibrium is thatthe total vector force and torque must vanish on the object(s) in question.Sounds simple!

But of course this is as many as six conditions, one for each of the possiblevector components of force and torque. All of which have to be zero at the sametime. Which means as many as six simultaneous equations per object have tobe satisfied. Urp.

Maybe not so simple?

287

288 CHAPTER 10. STATICS

10.1 Statics


Problem 243.

problems/statics-mc-elephant-mouse.tex

CM Elephant

CM mousem

m

1

L 1 L 2

2

pivot

An elephant and a mouse sit at either end of a really long, really strong see-saw.The elephant, whose mass is m1, sits so that its center of mass is a distanceL1 from the pivot. The mouse, whose mass is m2, sits at L2. The see-saw isbalanced so the mouse and elephant are not moving up or down. Which is thefollowing must be true:

a) m2 = m1

b) m2 = m1(L2/L1)

c) m1 = m2(L2/L1)

d) m1 = m2(L2/L1)2

e) The mouse can never balance the elephant!

10.1. STATICS 289

Problem 244.

problems/statics-mc-hanging-rope.tex

a

gravityM,L

b

In the figure above, a rope of mass M , length L is hanging from the ceiling instatic equilibrium. Select the correct rank order of the tension in the rope atthe points a and b:

a) Ta < Tb

b) Tb < Ta

c) Ta = Tb

d) Insufficient information given to determine the answer.


Problem 245.

problems/statics-mc-leaning-bar-reaction-pairs.tex

m

In the figure above, a board is sitting on a rough floor and leaning against awall. Circle three action-reaction pairs in the list below:

a) The ladder top pushes against the wall; the wall pushes back against theladder top.

b) The floor pushes up on the ladder base; gravity pulls the ladder base downtowards the floor.

c) Static friction from the floor pushes the ladder base towards the wall; thewall pushes back on the ladder.

d) The floor pushes down on the ground; the ground pushes back on thefloor.

e) The Earth pulls down on the ladder via gravity; the ladder pulls up onthe Earth via gravity.

10.1. STATICS 291

Problem 246.

problems/statics-mc-pick-reaction-pairs-2.tex

Which of the following list are not action-reaction force pairs? (More than oneanswer is possible.)

a) A hydraulic piston pushes on the fluid in its cylinder; the fluid pushesback on the hydraulic piston.

b) The earth’s gravity pulls a pendulum bob at rest down; the string pulls itup.

c) My finger pushes down against a grape I’m squeezing; my thumb pushesup against the grape.

d) My hammer pushes on a nail as it hits it; the nail pushes back on thehammer.

e) A bathroom scale pushes up on my feet as I stand on it; my feet pushdown on the scale.


Problem 247.

problems/statics-mc-pick-reaction-pairs.tex

(3 points) Which of the following list are not action-reaction force pairs? (Morethan one answer is possible.)

a) The earth’s gravity pulls down on an apple; the stem of the apple holds itup.

b) Water pressure pushes out against a glass, the glass holds in the water.

c) I push forward on a bow; the bowstring pulls forward on me (as I drawan arrow).

d) I lean my head on the wall; the wall pushes back on my head.

e) I pull down on the rope with my hand; the rope pulls up on my hand.

10.1. STATICS 293

Problem 248.

problems/statics-mc-plank-mass-rod.tex

mm

mm

(3 points) In the figure above, a very light (approximately massless) plank sup-ports a mass m. The plank is resting on (not attached to) a sawhorse that cansupport as much weight as you like, and a rod is attached to the plank as shown(where the other end is firmly attached to the ceiling or floor as the case maybe). The rod, however, will break if it is compressed or stretched with a forceFb = mg, the weight of the mass.

Circle all of the configurations where the plank and mass will not move and therod will not break.


Problem 249.

problems/statics-mc-plank-mass-string.tex

m m

mm

In the figure above, a very light (approximately massless) plank supports a massm. The plank is resting on (not attached to) a sawhorse/pivot that can supportas much weight as you like, and a massless string is attached to the plank asshown (the other end is tied to the ceiling or floor as the case may be). Thestring, however, will break at a force Fb = mg, the weight of the mass.

Circle all of the configurations where the plank and mass will not move and thestring will not break.

10.1. STATICS 295

Problem 250.

problems/statics-mc-string-and-bar.tex

ML

T30

o

A bar of mass M and length L is pivoted by a hinge on the left and is supportedon the right by a string attached to the wall and the right hand end of the bar.The angle made by the string with the bar is θ = 30. Select the true statementfrom the list below.

a) T = Mg/2

b) T = Mg

c) T =√

32 Mg

d) T = 2Mg

e) There is not enough information to determine T .


Problem 251.

problems/statics-mc-support-the-picture.tex

θθ

m

A picture of mass m has been hung by a piece of thread as shown. The threadwill break at a tension of mg. Find the smallest angle theta such that the threadwill not break. FYI: sin(30) = cos(60) = 1/2, cos(30) = sin(60) =

√3/2,

sin(45) = cos(45) =√

2/2, sin(90) = cos(0) = 1.

a) 30

b) 45

c) 60

d) 90

10.1. STATICS 297

Problem 252.

problems/statics-mc-tipping-blocks.tex

A B C D

In the figure, four blocks are placed on an inclined plane that has sufficientstatic friction that the blocks will not slip. The dots in the figures indicate thecenter of mass of each block. Which of the following is/are true?

a) A and D will tip.

b) A B and D will not tip.

c) B and C will tip.

d) C and D will tip.


Problem 253.

problems/statics-mc-torque-direction-block-held-to-wall-friction.tex

FP g

A cube of mass M is held at rest against a vertical rough wall by applyinga perfectly horizontal force ~F as shown. Gravity is down as usual as shown.What is the direction of the torque about the point P due to the force offriction exerted by the wall on the block?

a) Left.

b) Right.

c) Up.

d) Down.

e) Into the plane of the figure.

f) Out of the plane of the figure.

g) The torque is zero, so the direction is undefined.

10.1. STATICS 299

Problem 254.

problems/statics-mc-torque-direction-block-held-to-wall-normal.tex

FPg

A cube of mass M is held at rest against a vertical rough wall by applyinga perfectly horizontal force ~F as shown. Gravity is down as usual as shown.What is the direction of the torque about the point P due to the normalforce exerted by the wall on the block?

a) Left.

b) Right.

c) Up.

d) Down.





Problem 255.

problems/statics-mc-torque-direction-block-held-to-wall.tex

FP g

A cube of mass M is held at rest against a vertical rough wall by applyinga perfectly horizontal force ~F as shown. Gravity is down as usual as shown.What is the direction of the torque about the point P due to the force offriction exerted by the wall on the block?

a) Left.

b) Right.

c) Up.

d) Down.




Now, what is the direction of the torque about the point P due to the normalforce exerted by the wall on the block?

a) Left.

b) Right.

c) Up.

d) Down.




10.1. STATICS 301


Problem 256.

problems/statics-ra-board-and-pivot-force.tex

b

c

a

In the three figures above, a massless board is held in static equilibrium bya hinge at the left end and a trestle. A mass M is placed on the board at thethree places shown. For each figure:

a) Draw an arrow at the hinge indicating the direction of the force (if any)exerted by the hinge for all three figures. If the force is zero please indicatethis.

b) Rank the three figures in the order of the magnitude of the force exertedon the board by the trestle, from least to greatest.


Problem 257.

problems/statics-ra-chain-of-hanging-monkeys.tex

5md

c

b

a

m

m

m

m

In the figure above four monkeys, each of mass m, are shown holding very stillas they hang from a pole at the top of a circus tent. The top monkey (a)is holding a strap attached to the pole above, and the bottom monkey (d) isholding a mass 5m above with his foot. Which monkey (a-d) is pulling up withthe largest force with its feet?

10.1. STATICS 303

Problem 258.

problems/statics-ra-holding-up-the-bar.tex

Fc

Fb

Fa

A bar of mass M is pivoted by a hinge on the left and has a wire attached to theright as shown. The wire can be attached to the ceiling on eyebolts on any oneof the three angles shown to suspend the rod so that it is in static equilibrium.Rank the force F exerted on the rod by the wire when the wire comes offin the a, b, c directions (where equality is a possibility). That is, your answermight look like Fa < Fb = Fc (but don’t count on this being the answer). Notewell: The arrows in the figure above are not proportional to the forces, theyindicate only the directions.


Problem 259.

problems/statics-ra-stack-of-standing-monkeys.tex

5mm

m

m

m

b

c

d

a

(3 points) In the figure above four monkeys, each of mass m, are shown holdingvery still in a tower they’ve made at the circus. The bottom monkey (d) isstanding on the floor, the top monkey (a) is holding a mass 5m above his head.Which monkey is pushing up with the largest force with its arms?

10.1. STATICS 305

Problem 260.

problems/statics-ra-tipping-shapes-2.tex

h h

hh

A B

C D

θ

In the four figures above, the coefficient of static friction is high enough that theuniform objects shown will not slip before they tip. Rank the angles at whicheach mass will tip over as the right end of the plank they sit on is raised, fromsmallest (tipping) angle to the largest (for example, A,B,C,D).


Problem 261.

problems/statics-ra-tipping-shapes.tex

In the figure above, three shapes (with uniform mass distribution and thickness)are drawn sitting on a plane that can be tipped up gradually. Assuming thatstatic friction is great enough that all of these shapes will tip over before theyslide, rank them in the order they will tip over as the angle of the board theyare sitting on is increased. Be sure to indicate any ties.

10.1. STATICS 307

10.1.3 Short Answer

Problem 262.

problems/statics-sa-balance-the-mobile-2.tex

d3d

d 2d

M M B

M A

A static mobile suspends three patterned blocks over a baby’s bed. The lengthsof the supporting rigid rods (of negligible mass) are given in the figure above,as is the mass of the central block, M . Find MA and MB in terms of M sothat the mobile perfectly balances. Note well that the unknown blocks are notnecessarily drawn to scale!

a) MA =

b) MB =


Problem 263.

problems/statics-sa-balance-the-mobile-reversed.tex

M 3

d

d

x

y

M 21M

A static mobile suspends three patterned blocks over a baby’s bed. The massesof the blocks and the lengths of the supporting rigid rods (of negligible mass) aregiven in the figure above (although the relative distances may not be correctlyto scale). Find x and y in terms of d so that the mobile perfectly balances when:

M1 = 1 kg, M2 = 3 kg, M3 = 1 kg

x =

y =

10.1. STATICS 309

Problem 264.

problems/statics-sa-balance-the-mobile.tex

x d

yd

M

MM

1

23

A static mobile suspends three patterned blocks over a baby’s bed. The massesof the blocks and the lengths of the supporting rigid rods (of negligible mass)are given in the figure above. Find x and y in terms of d so that the mobileperfectly balances when:

M1 = 1 kg, M2 = 4 kg, M3 = 1 kg

x =

y =


Problem 265.

problems/statics-sa-leaning-bar-reaction-pairs.tex

m

In the figure above, a board is sitting on a rough floor and leaning against awall. Identify three action-reaction force pairs in the figure.

10.1. STATICS 311

Problem 266.

problems/statics-sa-pendulum-bob.tex

4 N

θ

F = 3 N

A 4 N pendulum bob supported by a massless string is held motionless at anangle θ from the vertical by a horizontal force F = 3 N as shown. The stringused to hang the mass will break at any tension T > Tc = 4

√2 N.

a) What is the angle θ (expression OK).

b) The force F is slowly increased. At what value will the string break?

c) What is the angle θ at which the string breaks?


Problem 267.

problems/statics-sa-suspended-food-bag.tex

m

θ θ

F

A gold prospector living in a rustic cabin mounts a sturdy wooden peg and three(approximately massless and frictionless) pulleys in fixed positions on the walland rafters as shown in the diagram so he can suspend his food bag up off thefloor and away from mice. He hangs a bag of food of mass m so that the ropemakes an angle θ with the central pulley as shown.

Help him find the magnitude of the force F that his rafter must exert downwardon the pulley when he has hung his bag of food.

10.1. STATICS 313

Problem 268.

problems/statics-sa-which-mass-breaks-string.tex

m

m

m

C)

B)

A)

L/3

2L/3

2L/3 T

T

T

L

L

L

y

y

y

x

x

x

In the figure above, a massless plank supports a massive block m placed at thelocations shown. The plank is supported by a wedge shaped support and astring that will break at the same tension Tmax (in all three cases) positionedas shown.

a) Suppose the mass m is gradually increased (in all three figures). In whichconfiguration (A, B, or C) will the string break first?

b) For that configuration (that you picked in part a), what is the value ofthe upward support force Fs exerted by the wedge right as (just before)the string breaks?


10.1.4 Long Problems

Problem 269.

problems/statics-pr-arm-with-barbell.tex

M

F?

D

m

d

θ

T?


• Static Equilibrium

• Force and Torque


An exercising human person holds their arm of mass M and a barbell of massm at rest at an angle θ with respect to the horizontal in an isometric curl asshown. The muscle that supports the suspended weight is connected a shortdistance d up from the elbow joint. The bone that supports the weight haslength D.

a) Find the tension T in the muscle, assuming for the moment that the centerof mass of the forearm is in the middle at D/2. Note that it is much larger

than the weight of the arm and barbell combined, assuming a reasonableratio of D/d ≈ 25 or thereabouts.

b) Find the force ~F (magnitude and direction) exerted on the supportingbone by the elbow joint. Again, note that it is much larger than “just”the weight being supported.

10.1. STATICS 315

Please note that this is not an accurate description of musculature – in actualfact several muscles would work together to achieve the desired static equi-librium (or to do the dynamic work of actual barbell curls). However, all ofthese muscles work with short moment arms at the point of bone attachment tosupport weights with long moment arms and the idea and methodology of thecalculation of the stresses on the muscles and joints is just more complicatedversions of this same reasoning. Sports medicine and orthopedic medicine andprosthetic design all use this sort of physics every day.


Problem 270.

problems/statics-pr-bar-and-pulleys.tex

F?

m

2L/3

L/3θ

m1?

M

Find the components of the pivot force ~F = (Fx, Fy) and find m1 in termsof M and m as givens in the figure above, if the bar of mass m is in staticequilibrium.

10.1. STATICS 317

Problem 271.

problems/statics-pr-bear-seeking-goodies.tex

A bear of mass MB walks out on a beam of mass mb to get a basket of foodof mass of mass mf . The beam has length L, and is supported by a wire at anangle of 60 degrees, as in the sketch.

a) Draw a free-body diagram for the beam that shows all forces. Includean indication of a coordinate system and also indicate the origin of thatcoordinate system.

b) Using that origin (as a pivot), write down all the force and torque balanceequations, assuming that the bear is located a distance x from the left endof the beam.

c) Solve these equations to find the (vector) force that the wall exerts on theleft end of the beam.

d) Find the tension in the wire.

e) Suppose that the bear is too heavy to reach the basket without breakingthe wire. If the maximum tension that the wire can support withoutbreaking is Tmax, find an expression for the largest distance from the wallxmax that the bear can walk without breaking the wire.


Problem 272.

problems/statics-pr-brace-against-house.tex

L

θ

Fpivot

M

2L/3

N

In the figure above, a “massless” rigid beam of length L that makes an angle ofθ with the ground is leaned against a frictionless wall at the upper end, whichexerts a normal force only N as shown on the beam. A mass M is suspendedvertically from a point 2/3 of the way from the pivot attached to the ground.Find:

a) The magnitude of the normal force N exerted by the wall on the beamwhen the entire beam is in static equilibrium.

b) The vector force ~F p exerted by the pivot on the ground on the beam tohold the beam in place. It is probably easiest to express this answer asFpx and Fpy.

10.1. STATICS 319

Problem 273.

problems/statics-pr-crane-boom.tex

boom

30 45

L

F?

mT?

M

A crane with a boom (the long support between the body and the load) of massm and length L holds a mass M suspended as shown. Assume that the centerof mass of the boom is at L/2. Note that the wire with the tension T is fixedto the top of the boom, not run over a pulley to the mass M .

a) Find the tension in the wire.

b) Find the force exerted on the boom by the crane body.

Note:

sin(30) = cos(60) =1

2

cos(30) = sin(60) =

√3

2

sin(45) = cos(45) =

√2

2


Problem 274.

problems/statics-pr-crane-vertical-support.tex

M

L/4

L

θ

Fpivot

F

In the figure above, a “massless” rigid beam of length L that makes an angle ofθ with the ground is braced with a piece of wood a distance L/4 from the endon the ground. This piece of wood is attached at right angles to the beam asshown. At the upper end of the beam a mass M is suspended. Find:

a) The magnitude F of the force exerted by the support bar when theentire beam is in static equilibrium.

b) The vector force ~F p exerted by the pivot on the ground on the beam(not the support bar) to hold the beam in place. It is probably easiest toexpress this answer as Fpx and Fpy .

10.1. STATICS 321

Problem 275.

problems/statics-pr-cylinder-and-corner-2.tex

h

R

MF

A cylinder of mass M and radius R sits against a step of height h = R/2 as

shown above. A force ~F is applied parallel to the ground as shown. All answersshould be in terms of M , R, g.

a) Find the minimum value of |~F | that will roll the cylinder over the step ifthe cylinder does not slide on the corner.

b) What is the force exerted by the corner (magnitude and direction) when

that force ~F is being exerted on the center?


Problem 276.

problems/statics-pr-cylinder-and-corner.tex

h

R

F

M



• Torque (about selected pivots)

• Geometry of Right Triangles


A cylinder of mass M and radius R sits against a step of height h = R/2 as

shown above. A force ~F is applied at right angles to the line connecting thecorner of the step and the center of the cylinder. All answers should be in termsof M , R, g.

a) Find the minimum value of |~F | that will roll the cylinder over the step ifthe cylinder does not slide on the corner.

b) What is the force exerted by the corner (magnitude and direction) when

that force ~F is being exerted on the center?

10.1. STATICS 323

Problem 277.

problems/statics-pr-dangling-bar.tex

mθ L/4

LT

2m

In the figure above, a rod of length L with mass m is suspended by a hinge onthe left and a horizontal string on the right. A second mass 2m is suspendedfrom the rod a distance L/4 from the hinge end. Find:

a) The tension T in the horizontal string.

b) The vector force ~F exerted by the hinge, in any of the acceptable formswe use to completely specify a vector.


Problem 278.

problems/statics-pr-disk-in-corner.tex

M

R

θ

θ

a

b

Find the magnitude of the normal forces Na and Nb exerted by the two wallson the disk of mass M and radius R at the points a and b such that it sits instatic equilibrium in the picture above:

• Na =

• Nb =

10.1. STATICS 325

Problem 279.

problems/statics-pr-double-diagonal-pulleys.tex

θ

M

F?

In the figure above, two massless pulleys and a massless unstretchable stringsupport a mass M in static equilibrium as shown. The pulleys are fixed onunmoveable frictionless axles.

a) (3 points) Draw a force diagram for the mass M and both pulleys.

b) (5 points) Find the vector force ~F exerted by the axle of the upper pulleyat equilbrium.

c) (1 point) If the angle θ is increased (by lowering the lower pulley, forexample) is there more or less force exerted by the upper axle to keep thepulley in place?


Problem 280.

problems/statics-pr-floating-buoy.tex

water

T?

1 m3

A round buoy at the beach floats in fresh water when it is exactly half sub-merged. Its spherical volume is 1 cubic meter. If it is pulled all the way under-water and suspended from the bottom by means of an anchored rope, what isthe tension in the rope?

10.1. STATICS 327

Problem 281.

problems/statics-pr-hang-a-stable-T.tex

W

W

T

T

2

3

1

T

Mg/2

Mg/2

The “T” shaped object above has mass M , and has both a height and widthof W . Assume that this mass is uniformly distributed in the long arm andthe crossbar, that is, that the center of mass of the long arm is at W/2 and thecenter of mass of the crossbar is also at W/2 and that the long arm and crossbareach has mass M/2 (and hence gravity exerts a downward force at their centersof mass of Mg/2 as shown).

Find the tension T1,2,3 in each of the three ropes that support the T above.Note that the ropes all pull straight up (they are vertical) and the T is com-pletely horizontal.


Problem 282.

problems/statics-pr-hanging-ball-2.tex

mg

4

4

(9 points total) In the figure above, a mass m is hanging from two massless,unstretchable ropes. Gravity pulls straight down on the mass with a force ofmagnitude mg. Assume that the tension in both ropes has the equal magnitudeT . The mass is hanging 4 meters beneath the ceiling, and each rope is fastenedto the ceiling offset by 4 meters from where the mass hangs as shown.

a) (3 points) Draw a coordinate system and free body diagram representingall the forces acting on the hanging mass. Label any angles that might beof use to you.

b) (3 points) Write the algebraic equations for the total force in the x and ydirections that are the conditions for static equilibrium.

c) (3 points) Find the tension T in terms of mg.

10.1. STATICS 329

Problem 283.

problems/statics-pr-hanging-ball.tex

mg

4

5

In the figure above, a mass m is hanging from two massless, unstretchable ropes.Gravity pulls straight down on the mass with a force of magnitude mg. Assumethat the tension in both ropes has the equal magnitude T . The length of theeach rope is 5 meters, and the mass is hanging 4 meters beneath the ceiling asshown

a) Draw a coordinate system and free body diagram representing all theforces acting on the hanging mass. Label any angles that might be of useto you.

b) Write the algebraic equations for the total force in the x and y directionsthat are the conditions for static equilibrium.

c) Find the tension T in terms of mg.


Problem 284.

problems/statics-pr-hanging-door.tex

W

d

d

HM

F

Fb

t

A door of mass M that has height H and width W is hung from two hingeslocated a distance d from the top and bottom, respectively. Assuming that theweight of the door is equally distributed between the two hinges, find the totalforce (magnitude and direction) exerted by each hinge. (Neglect the mass ofthe doorknob. The force directions drawn for you are NOT likely to be corrector even close.)

10.1. STATICS 331

Problem 285.

problems/statics-pr-hanging-tavern-sign.tex

θ

and

for sale

Mechanic−Ale

Physics Beer

wallL

sign (mass m)

massless rod of length Lmassless wire

T?F?

In the figure above, a tavern sign belonging to a certain home-brewing physicsprofessor is shown suspended from the middle of a massless supporting rodof length L (at L/2). Find the tension in the (massless) wire, T , and the total

force exerted on the suspending rod by the wall, ~F , in terms of m, g, L, andθ.

Please indicate the coordinate system you are using on the figure and the loca-tion of the pivot point used, if any.


Problem 286.

problems/statics-pr-inclined-plane.tex

mM

T

θ

T


• Newton’s Third Law


• Fully Inelastic Collisions


In the inclined plane problem above all masses are at rest and the pulley andstring are both massless. Find the normal force exerted by the inclined planeon the mass M and the mass m required to keep the system in static balancein terms of M and θ.

10.1. STATICS 333

Problem 287.

problems/statics-pr-ladder-on-glacier.tex

m

L θ

An ultralight (assume massless) ladder of length L rests against a vertical blockof (frictionless) ice during a hazardous ascent of a glacier at an angle θ = 30 asdrawn. A mountaineer of mass m climbs the ladder. When the mountaineer isstanding at rest at the very top of the ladder and about to reach over the cliffedge, what is the net force exerted on the base of the ladder by the glacier?


Problem 288.

problems/statics-pr-ladder-on-wall.tex

µs

θ

Mm

h


• Torque Balance

• Force Balance


• Static Friction


In the figure above, a ladder of mass m and length L is leaning against a wall atan angle θ. A person of mass M begins to climb the ladder. The ladder sits onthe ground with a coefficient of static friction µs between the ground and theladder. The wall is frictionless – it exerts only a normal force on the ladder.

If the person climbs the ladder, find the height h where the ladder slips.

10.1. STATICS 335

Problem 289.

problems/statics-pr-mass-two-strings.tex

Fm

θ

A ball of mass m hangs from the ceiling on a massless string. A second masslessstring is attached to the ball and a force ~F is applied to it in the horizontaldirection so that the system remains in static equilibrium in the position shown,where θ is the angle between the first string and the vertical. Gravity acts downas usual. Each string can support a maximum tension Tmax = 2mg withoutbreaking.

a) If ~F is slowly increased while keeping its direction horizontal, which stringwill break first? Explain your reasoning.

b) Find the maximum value θmax that the hanging string can have whenthe system is in static equilibrium with both strings unbroken. (You mayexpress this angle as an inverse sine, cosine, or tangent if you wish – youdo not need a calculator.)

c) Find the force magnitude Fmax that produces the maximum angle θmax

in static equilibrium. Express this answer in terms of m and g.


Problem 290.

problems/statics-pr-pendulum-bob-pulley.tex

θ

M

m

A pendulum bob of mass m is attached both to the ceiling and to a mass Mhanging over a pulley by unstretchable massless strings as shown. The pulley isfixed on an unmoveable frictionless axle.

a) (3 points) Draw free body diagrams for both mass m and mass M .

b) (3 points) Find an expression for the angle θ at which the system is instatic equilibrium.

c) (3 points) Find the total tension T in the string connecting the pendulumbob to the ceiling.

10.1. STATICS 337

Problem 291.

problems/statics-pr-pushing-down-wall.tex

M

D

pivot

H

x

Tom is a hefty construction worker (mass M = 100 kilograms) with a good senseof balance who wants to push down a brick wall. The wall, however, is strongenough to withstand any horizontal push up to 2000 N and Tom can only exerta sideways equal to his weight with his muscles.

Fortunately, Tom has a perfectly rigid 4 × 4 beam (of negligible mass), andthere is a solid rock (that can withstand essentially any push) a distance D = 5meters from the wall to brace it on. Even more fortuitously, Tom has takenintroductory physics! He therefore cuts the beam to lean against the wall aheight H as shown and proceeds to walk up the beam towards the wall..

a) Assuming that the beam is frictionless where it presses against the wallwhat is the largest value of H that will permit him to knock down the wallif he walks to the end of the beam so that his horizontal distance x = D?

b) Suppose that he has cut the beam so that it rests a height H = 1 meterabove the ground against the wall. What is his horizontal position x whenthe beam knocks down the wall (if it does at all)?

c) Of course the beam is not frictionless where it rests against the wall. Doesthis fact mean that, for any given value of H , the wall is easier to knockdown (happens when he has walked a smaller horizontal distance x towardthe wall), harder to knock down (happens when he has walked a greaterhorizontal distance x), or just the same (it falls at the same horizontaldistance x) as it is without friction?


Problem 292.

problems/statics-pr-rod-mass-on-hinge.tex

T

m

M

L

P

θ

A small round mass M sits on the end of a rod of length L and mass m thatis attached to a wall with a hinge at point P . The rod is kept from falling bya thin (massless) string attached horizontally between the midpoint of the rodand the wall. The rod makes an angle θ with the ground. Find:

a) the tension T in the string;

b) the force ~F exerted by the hinge on the rod.

10.1. STATICS 339

Problem 293.

problems/statics-pr-supporting-a-disk.tex

M

R

R/2

R/2

Find the force exerted by each of the two rods supporting the disk of mass Mand radius R as shown. Note that the two triangles shown are both 30-60-90triangles with side opposite the small angle of R/2.


Problem 294.

problems/statics-pr-tipping-vs-slipping.tex

θ

µ

3cm

s = 2/34cm

M

A block of mass M with width 3 cm and height 4 cm sits on a rough plank.The coefficient of static friction between the plank and the block is µs = 2/3.The plank is slowly tipped up. Does the block slip first, or tip first?

10.1. STATICS 341

Problem 295.

problems/statics-pr-vector-torque-plexiglass-table.tex

w

w/2

d/3

w/3d

m

Top view


• Force Balance

• Torque Balance



The figure below shows a mass m placed on a table consisting of three narrowcylindrical legs at the positions shown with a light (presume massless) sheet ofPlexiglas placed on top. What is the vertical force exerted by the Plexiglas oneach leg when the mass is in the position shown?

Chapter 11

Fluids

Fluids have statics too! Water can sit still in a drinking glass, held in by normalforces exerted by the glass, held down by gravity, and internally held in placeby – water. Even air is static (when their is no wind.

But the really interesting things are what happens when fluids move. We barelyscratch the surface in this course – fluid dynamics is arguably one of the mostdifficult theories in all of physics, especially in the general, nonlinear, chaotic,turbulent regime. Which is where we, and a whole lot of everyday stuff, live.

Fluid statics and dynamics is once again useful to everybody from physiciansto engineers to physicists to physicians to boat captains to airline pilots tophysicians, to...

Did I mention that the human body, from one point of view is a big, walking,talking, thinking bag of water (complete with pumps and plumbing) with a fewvery important contaminants in it?

No?

343

344 CHAPTER 11. FLUIDS

11.1 Fluids


Problem 296.

problems/fluids-mc-boat-catches-fish.tex

?

?

?

I go fishing in a pond where there is a big, fat fish perfectly suspended bybuoyant forces in the water under the boat. I catch him and reel him in up intothe boat. As I do so, the level of the water in the pond will:

a) Rise a bit.

b) Fall a bit.

c) Remain unchanged.

d) Can’t tell from the information given (it depends, for example, on the kindof fish...).

11.1. FLUIDS 345

Problem 297.

problems/fluids-mc-boat-floats-wood.tex

?

?

?

A person stands in a boat floating on a pond and containing several pieces ofwood. He throws the wood out of the boat so that it floats on the surface ofthe pond. The water level of the pond will:

a) Rise a bit.

b) Fall a bit.


d) Can’t tell from the information given (it depends on, for example, theshape of the boat, the mass of the person, whether the pond is located onthe Earth or on Mars...).


Problem 298.

problems/fluids-mc-boat-lowers-anchor.tex

?

?

?

I go fishing in a pond and spot a big, fat fish in the water under the boat anddecide to anchor for a bit to try to catch it. As I lower the anchor into the water(so that it hangs suspended under the boat as shown) level of the water inthe pond will:

a) Rise a bit.

b) Fall a bit.


d) Can’t tell from the information given (it depends, for example, on whetherthe anchor is made of iron or lead...).

11.1. FLUIDS 347

Problem 299.

problems/fluids-mc-boat-releases-balloons.tex

?

?

?

The fish aren’t biting, so a person standing in a boat floating on a pond andinflates a bunch of helium balloons instead. Then an enormous fish jumps nearbyand he is so startled that he accidentally releases the balloons. As he does so,the water level of the pond will:

a) Rise a bit.

b) Fall a bit.


d) Can’t tell from the information given.

(Ignore the fish!)


Problem 300.

problems/fluids-mc-boat-sinks-rocks.tex

?

?

?

A person stands in a boat floating on a pond and containing several large, round,rocks. He throws the rocks out of the boat so that they sink to the bottom ofthe pond. The water level of the pond will:

a) Rise a bit.

b) Fall a bit.


d) Can’t tell from the information given (it depends on, for example, theshape of the boat, the mass of the person, whether the pond is located onthe Earth or on Mars...).

11.1. FLUIDS 349

Problem 301.

problems/fluids-mc-buoyant-boxes.tex

Two wooden boxes with the same shape but different density are held in thesame orientation beneath the surface of a large container of water. Box A has asmaller average density than box B. When the boxes are released, they accelerateup towards the surface. Which box has the greater acceleration when they areinitially released?

a) Box A.

b) Box B.

c) They are the same.

d) We cannot tell from the information given.


Problem 302.

problems/fluids-mc-density-of-stone-underwater.tex

m =10 kg

60 N

water

A stone of mass m = 10 kg (that weighs 100 Newtons in air) is hung from ascale and immersed in water. The scale reads 60 Newtons. What is the densityof the stone? (Use g = 10 m/sec2)

a) ρ = 1000 kg/m3

b) ρ = 4000 kg/m3

c) ρ = 6000 kg/m3

d) ρ = 1667 kg/m3

e) ρ = 2500 kg/m3

11.1. FLUIDS 351

Problem 303.

problems/fluids-mc-floating-mass-on-spring-different-rho.tex

m

k

ρ

D

g

A)

A

m

k D

B)

g

B

ρ/2

In the figures above, two identical springs (with spring constant k) are attachedto the bottoms of two identical containers filled with two different fluids withdensities (A) ρ and (B) ρ/2 respectively. Wooden blocks that would ordinarilyfloat are attached to these springs, which stretch out to total lengths DA andDB and suspend the blocks so that they are fully immersed as shown.


DA > DB DA < DB DA = DB


Problem 304.

problems/fluids-mc-floating-mass-on-spring-elevator.tex

m

k

ρ

A)

∆ x A

xeq

m

k

ρ

B)

∆ xB

xeq

a

In the figures above, two identical springs (with spring constant k) are attachedto the bottoms of two identical containers filled with water (density ρ). Atthe other end, the springs are attached to identical wooden blocks that wouldordinarily float on the water so that they are completely submerged.

The container on the left (A) is located at rest on the ground, and ∆xA is thetotal distance that its spring is stretched from its equilibrium length when theblock is stationary relative to container A. The container on the right (B) islocated on the floor of an elevator accelerating upwards with an accelerationa, and ∆xB is the total length that its spring is stretched from its equilibriumlength when the block is stationary relative to container B (accelerating upwardswith the elevator).

Circle the true statement:∆xA > ∆xB ∆xA < ∆xb ∆xA = ∆xB

11.1. FLUIDS 353

Problem 305.

problems/fluids-mc-floating-mass-on-spring-moon.tex

m

k

ρ

De

g

A)

m

k

ρ

Dm

g/6

B)

In the figures above, two identical springs (with spring constant k) are attachedto the bottoms of two identical containers filled with water (density ρ). Atthe other end, the springs are attached to identical wooden blocks that wouldordinarily float on the water so that they are completely submerged.

The apparatus on the left (A) is located on the Earth’s surface, where theacceleration due to gravity is g. The apparatus on the right (B) is located onthe moon, where the acceleration due to gravity is g/6. De is the total lengthof the stretched spring on the Earth, Dm is the total length of the stretchedspring on the moon.

Circle the true statement:De > Dm De < Dm De = Dm


Problem 306.

problems/fluids-mc-flow-constricted-pipe.tex

d d/2v2v1

Water flows at speed v1 in a pipe with diameter d and passes into a pipe withdiameter d/2 through a smooth constriction as shown. Select the statementthat correctly describes v2, the speed in the narrower pipe.

a) v2 = 4v1

b) v2 = 2v1

c) v2 = v1

d) v2 = 12v1

e) v2 = 14v1

11.1. FLUIDS 355


Problem 307.

problems/fluids-ra-archimedes-objects-2.tex

Four large identical beakers are filled with water and also contain the followingobjects in order of decreasing density :

a) A solid gold coin that has a mass of 100 grams.

b) A cast aluminum frog that has a mass of 100 grams.

c) An ice cube that has a mass of 100 grams.

d) A wooden carved monkey that has a mass of 100 grams.

These objects are not attached to anything, they are in static equilibrium withthe water. You remove the objects from the water in each beaker and measurethe drop in water depth ∆di, i = a, b, c, d.

Rank the ∆di you expect to observe in this experiment from smallest tolargest. As always, in the case that some of the ∆di are equal to neighbors,indicate that explicitly.


Problem 308.

problems/fluids-ra-archimedes-objects.tex

A large beaker is filled to a marked line with water. You have the followingobjects (in order of decreasing density):

a) A solid gold coin that has a mass of 100 grams.

b) A cast aluminum frog that has a mass of 100 grams.

c) An ice cube that has a mass of 100 grams

d) A 100 gram chunk of shipping styrofoam.

You drop each item, one at a time, into the beaker in the water and record di,the change in water depth, and then remove it.

Rank the expected results for di for i = a, b, c, d. Indicate whether di ispositive (so that the water in the beaker rises) or negative (falls). As always, inthe case that some of the di are equal to neighbors, indicate that explicitly.

11.1. FLUIDS 357

Problem 309.

problems/fluids-ra-four-utubes-venturi.tex

A DC

air air

B

In the four u-tubes pictured above, only one of the two cases in each pair (Avs B and C vs D) make sense. In A vs B, a can of compressed air is blowingair across the top of one of the tube tops and the tube contains only a singlefluid. In C vs D, the density of the immiscible fluids is indicated by the shadingwhere the darker fluid has the greater density.

Which two u-tubes DO make physical sense? (Circle one of eachpair.)

A B C D


Problem 310.

problems/fluids-ra-poiseiulles-law-2.tex

2L

2LL

L

A

C

B

D

2r

rr

2r

In the figure above, several circular pipes carry fluids with the same viscosity.Rank the pipes in the order of their resistance to laminar flow, from least togreatest. Equality is a possible answer. Think carefully about the dependence onr in Poiseuille’s Law! This is why obstructions in arteries increase the resistanceso dramatically!

11.1. FLUIDS 359

Problem 311.

problems/fluids-ra-poiseiulles-law.tex

Pright

Pright

Pleft

Pleft

Pleft

Pright

Pright

Pleft

r

2Lr

D

C

B

A

2L

L

L

I

I

I

I D

C

B

A

2r

2r

Poiseuille’s Equation is given by:

∆P = Pleft − Pright =8µL

πr4I

Rank the volume flow I from the lowest to highest in the boxes below using“<” or “=” signs in between for the four circular pipes illustrated in the figureabove, assuming that in all cases ∆P is the same and the same fluid (with thesame viscosity µ) is flowing through the pipes.


Problem 312.

problems/fluids-ra-three-crowns.tex

A B C

Three crowns are shown above. Crown A is made of solid lead (specific gravity11.3) covered with a thin veneer of gold leaf. Crown B is made of platinum(specific gravity 21.5), also covered with a thin veneer of gold leaf. Crown C ismade of pure gold (specific gravity 19.3). All three crowns weigh exactly 500grams in air. Rank the crowns in the order of effective weight while immersedin the water (what the three scales will read) lowest to highest.

11.1. FLUIDS 361

11.1.3 Short Answer

Problem 313.

problems/fluids-sa-aneurism-pressure-flow.tex

C

B

A vn

vn

vn

nP

Normal

Obstruction

Aneurism

Consider the models above of a normal blood vessel (A), an obstructed bloodvessel (B) and an aneurism (C). In case (A) blood is flowing from left to right ata “normal” fluid velocity vn and pressure Pn. vn must be the same in the partof the vessel of the same cross-sectional area in (B) and (C) as well in order tomaintain adequate perfusion of the tissue supplied by these two vessels. Answerthe following questions briefly explaining the physics behind your answer (ornaming the rule, effect, principle, involved):

a) Is the blood pressure in the obstructed region in (B) higher or lower thannormal?

b) Is the blood pressure in the aneurism itself in (C) higher or lower thannormal?

c) Is the blood pressure in the vessel immediately before and after the ob-structed region in (B) higher or lower than normal (in order to maintainadequate flow)?


Problem 314.

problems/fluids-sa-balloon-in-car.tex

A small boy is riding in a minivan with the windows closed, holding a heliumballoon. The van goes around a corner to the left. Does the balloon swing tothe left, still pull straight up, or swing to the right as the van swings aroundthe corner?

11.1. FLUIDS 363

Problem 315.

problems/fluids-sa-breathing-underwater-through-a-tube.tex

In adventure movies, the hero is often being chased by the bad guys and escapesby hiding deep underwater and breathing through a tube of some sort. Assumingthat you can barely manage to breathe if a 500 Newton person is standingdirectly on your chest while you are lying on the floor, estimate the maximumdepth (of your chest) where one is likely to have the muscular strength to beable to breathe through a rigid tube extending to the surface. Your estimateshould be quantitative and you should support it with both a very short pieceof algebra and a picture clearly showing the forces you must work against tobreathe underwater.


Problem 316.

problems/fluids-sa-four-utubes-1.tex

b da c

Two different incompressible fluids separated by a thin (massless, frictionless)piston so that they cannot mix are open to the atmosphere and are in staticequilibrium in each of the four U-tubes pictured above.

a) One of the four U-tubes makes no sense (cannot be in equilibrium).Circle it and label it ”impossible”.

b) Underneath each u-tube that does make sense indicate whether the fluidat the top of the left-hand side of the “U” is denser than, less densethan, or the same density as the fluid at the top of the right-handside.

As always, briefly indicate your reasons.

11.1. FLUIDS 365

Problem 317.

problems/fluids-sa-four-utubes-2.tex

b da c

(6 points) Two different incompressible fluids separated by a thin (massless,frictionless) piston so that they cannot mix are open to the atmosphere andpresumably in static equilibrium in each of the four u-tubes pictured above.One of the four u-tubes makes no sense (cannot be in equilibrium). Circle it.Underneath each u-tube that does make sense indicate whether the fluid at thetop of the left-hand side of the “u” is denser than, less dense than, or the samedensity as the fluid at the top of the right-hand side. Briefly indicate yourreasons.


Problem 318.

problems/fluids-sa-hydraulic-lift-2.tex

A a

Fluid

fM

A piston of small cross sectional area a is used in a hydraulic press to exert aforce f on the enclosed liquid. A connecting pipe leads to the larger piston ofcross sectional area A, so that A > a. The two pistons are at the same height.The weight w = Mg that can be supported by the larger piston is

(a) w > f

(b) w < f

(c) w = f

(d) depends on whether the liquid is compressible or not.

11.1. FLUIDS 367

Problem 319.

problems/fluids-sa-hydraulic-lift.tex

M

A a

m

Fluid

The pair of coupled piston-and-cylinders shown above are sitting in air andfilled with an incompressible fluid. The entire system is in static equilibrium(so nothing moves). The cross-sectional area of the large piston is A; the cross-sectional area of the small piston is a. In this case we know that:

a) M = Aa m

b) M = aAm

c) M =√

Aa m

d) M = m

e) We cannot tell what M is relative to m without more information.


Problem 320.

problems/fluids-sa-poiseiulles-law-2.tex

Resistance RA

2Lr

D

C

B

2L

L

rL

rAL

E

η

η

η

η

2η

2r

2r

In the figure above, fluids of the given viscosities flow through circular pipesA-E with the given dimensions. The resistance to fluid flow of circular pipeA is known to be RA. What are the resistances of the other four pipes in termsof RA?

RB

RA=

RC

RA=

RD

RA=

RE

RA=

11.1. FLUIDS 369

Problem 321.

problems/fluids-sa-poiseiulles-law-3.tex

Resistance RA

2Lr

D

C

B

2L

L

rL

rAL

E

η

η

η

η

2η

Q

Q

Q

Q

Q

lPhP

2r

2r

In the figure above, fluids of the given viscosities flow through circular pipes A-Ewith the given dimensions. In all cases the volumetric flow through the pipesis held constant at Q by varying the pressure difference ∆Pi = Phigh − Plow

across each (i = A, B, C, D, E) pictured pipe segment.

The pressure difference that maintains flow Q fluid flow of circular pipe A isdefined to be ∆PA. What are the pressure differences across the other fourpipes in terms of ∆PA?

∆PB

∆PA=

∆PC

∆PA=

∆PD

∆PA=

∆PE

∆PA=


Problem 322.

problems/fluids-sa-poiseiulles-law.tex

Pleft Pright

Pleft

Pleft

Pleft

Pright

Pright

Pright

c

I v

I v

I v

I v

a

b

d

L

r

Use Poiseuille’s Law to answer the following questions:

a) Is ∆Pa = Pleft−Pright greater than, less than, or equal to zero in figure a)above, where blood flows at a rate Iv horizontally through a blood vesselwith constant radius r and some length L against the resistance of thatvessel?

b) If the radius r increases (while flow Iv and length L remain the same asin a), does the pressure difference ∆Pb increase, decrease, or remain thesame compared to ∆Pa?

c) If the length increases (while flow Iv and radius r remains the same asin a), does the pressure difference ∆Pc increase, decrease, or remain thesame compared to ∆Pa?

d) If the viscosity η of the blood increases (where flow Iv, radius r, andlength L are all unchanged compared to a) do you expect the pressuredifference ∆Pd difference across a blood vessel to increase, decrease, orremain the same compared to ∆Pa?

11.1. FLUIDS 371

Problem 323.

problems/fluids-sa-siphon.tex

H

A siphon is a device for lifting water out of one (higher) reservoir and deliveringit another (lower) reservoir as shown above. Estimate the probable maximumheight H one can lift the water above the upper reservoir’s water level beforethe tube descends into the lower reservoir. Explain your reasoning – how, andwhere, will the siphon fail?


Problem 324.

problems/fluids-sa-utube-fluid-height.tex

yR

yL


• Pascal’s Principle



A vertical U-tube open to the air at the top is filled with oil (density ρo) onone side and water (density ρw) on the other, where ρo < ρw. Find yL, theheight of the column on the left, in terms of the densities, g, and yR as needed.Clearly label the oil and the water in the diagram below and show all reasoningincluding the basic principle(s) upon which your answer is based.

11.1. FLUIDS 373

Problem 325.

problems/fluids-sa-walking-in-a-pool.tex

People with vascular disease or varicose veins (a disorder where the veins inone’s lower extremeties become swollen and distended with fluid) are often toldto walk in water 1-1.5 meters deep. Explain why.



Problem 326.

problems/fluids-pr-bernoulli-beer-keg-horizontal.tex

P

H

h

R

P0

BEER

A

a


• Bernoulli’s Equation

• Toricelli’s Law


In the figure above, a CO2 cartridge is used to maintain a pressure P on topof the beer in a beer keg, which is full up to a height H above the tap at thebottom (which is obviously open to normal air pressure) a height h above theground. The keg has a cross-sectional area A at the top. Somebody has pulledthe tube and valve off of the tap (which has a cross sectional area of a) at thebottom.

a) Find the speed with which the beer emerges from the tap. You may usethe approximation A ≫ a, but please do so only at the end. Assumelaminar flow and no resistance.

b) Find the value of R at which you should place a pitcher (initially) to catchthe beer.

11.1. FLUIDS 375

c) Evaluate the answers to a) and b) for A = 0.25 m2, P = 2 atmospheres,a = 0.25 cm2, H = 50 cm, h = 1 meter and ρbeer = 1000 kg/m3 (the sameas water).


Problem 327.

problems/fluids-pr-bernoulli-constricted-pipe.tex

d d/2

1P P2

v2v1

Water flows at a pressure P1 and a speed v1 in a circular storm culvert pipeof diameter d. The pipe narrows smoothly to a second pipe section where thediameter is only d/2.

a) Find v2, the speed in the second pipe.

b) Find P2, the pressure in the second pipe.

c) Write an algebraic expression in terms of the givens for the current (flow)I, the volume of water per second that passes through the pipe(s).

d) Evaluate your answer(s) given the data: P1 = 1.075 × 105 Pa, v1 = 1m/sec, d = 2/

√π meters. No calculator should be needed.

11.1. FLUIDS 377

Problem 328.

problems/fluids-pr-bernoulli-constricted-pipe-vertical.tex

d

H

d/2

air pressure

v1

2v

A drain pipe in a house starts out at a diameter of d and narrows smoothly toa second pipe section where the diameter is only d/2. It is filled with water toa height H above the exit point of the lower pipe where it empties into a stormsewer. Both ends of the pipe are open to the air.

a) Find v1 and v2, the speed of the flowing fluid in both pipe sections.

b) Write an algebraic expression in terms of the givens for the current (flow)Q, the volume of water per second that passes through the pipe(s). Givethe expression in terms of d and v1 and/or v2 so that your answer doesnot depend on your answer to a).

c) How long ∆t will it take for the water level in the top pipe to drop adistance ∆x ≪ H?


Problem 329.

problems/fluids-pr-bernoulli-emptying-iced-tea-time.tex

y

A

A 1

v1

v2

2

In the figure above, a big jug of iced tea with cross-sectional area A1 is open tothe air on top. A tap on the bottom has a hole with a cross-sectional area A2.The surface of the iced tea is a height H above the tap.

a) Find the rate at which the height of the iced tea drops – dH/dt – whenthe tap is opened.

b) How long does it take for all the iced tea to run out?

You may assume that A1 ≫ A2 and use any approximations that may suggest.

11.1. FLUIDS 379

Problem 330.

problems/fluids-pr-bernoulli-irrigation-pipe.tex

P

1

2

P0

H

WATER

In the figure above, a pump maintains a pressure of P in the air at the top ofa tank of water with a cross sectional area A. An irrigation pipe at the bottomleads up a slope to a farmer’s field. The vertical distance between the top surfaceof water in the tank and the opening of the pipe is H . The cross-sectional areaof the pipe is a. The top pipe is open to air pressure P0 = 1 atm. Recall thatthe density of water is ρ = 103 kg/m3.

a) What is the velocity of the water coming from the pipe? (Find this alge-braically from the appropriate law(s).)

b) Is the pressure at the bottom of the tank greater inside the main vessel(point 1 on figure above) or inside the pipe (point 2)? Briefly explain.

c) After finding the answer to a) algebraically and answering b), evaluate vnumerically using: P = 2.5 atm, A = 10 m2, H = 10 m, and a = 4 cm2.You shouldn’t need a calculator for this.


Problem 331.

problems/fluids-pr-bernoulli-IV.tex

A

1

2

a

P

PH

vein

Although mechanized and precise in modern first-world medicine, IV fluid de-livery in the rest of the world is an imprecise gravity-driven system. A bag orbottle filled with a saline solution, plasma, blood, or medicine is hung abovea patient’s bed and a tube delivers that fluid directly into a patient’s vein. Aphysician practicing medicine in many clinics or hospitals around the world maywell need to be able to estimate things like the time of delivery of a bolus offluid by a gravity-driven IV line for a given needle size.

Make such an estimate below, assuming that the bag of cross-sectional area Aholds a fluid of density ρ, is effectively open to air pressure in the room P1, andis suspended a height H above the level of the patient as shown. Use a as thecross-sectional area of the needle. Ignore viscosity and the fluid flow resistanceof the tubing. Express all your answers algebraically in terms of A, a,ρ, P1 and P2 for full credit.

a) What is the minimum height Hmin such that flow is from the bag to thepatient instead of from the patient back towards the bag? (We don’t wantthe patient to inadvertently donate blood!)

b) Suppose you raise the bag height to H = 2Hmin. With what velocity doesthe fluid flow into the patient?

c) If the bag holds a fluid volume V estimate how long does it will taketo deliver all of the fluid in the bag into the patient at this new height.Assume that H does not change (much) while the bag empties.

d) If one included viscosity and the drop in fluid height as the bag empties,would it increase or decrease the time from this rough estimate?

After finding the algebraic answers, you may estimate the numerical valuesof these quantities without a calculator for one point of EXTRA credit per

11.1. FLUIDS 381

answer for a maximum of three extra points. Assume that the fluid is water,V = 500 cubic centimeters, P1 = 1 atm, P2 = 1.1 atm, A = 2 × 10−3 m2, anda =

√5× 10−7 m2. (Note that leaving radicals like

√5 in your answers is OK).


Problem 332.

problems/fluids-pr-bernoulli-range-arc.tex

x

H

y

θ

va

R

P

Pt

a

A sealed tank of water (density ρ) is shown above. Inside it is pressurizedto a pressure Pt = 3Pa (where Pa is the pressure outside of the tank, oneatmosphere). The water escapes through a small pipe at the bottom wherethe stream is angled up at an angle θ with respect to the ground as shown. Thecross-sectional area of the tank A is much larger than the cross-sectional area aof the small pipe at the bottom, A ≫ a. (Picture is not necessarily to scale.)

a) What is the (approximate) speed va with which the water exits the smallpipe? Express your answer (for this part only) in terms of ρ, g, Pt, Pa andpossibly A and a.

b) What is the horizontal range of the stream of water, R, measured fromthe tip of the spout as shown. Express your answer (for this part only) interms of va.

11.1. FLUIDS 383

Problem 333.

problems/fluids-pr-bernoulli-range-horizontal.tex

H

y

xR

va

h

Pt

Pa

A sealed tank of water (density ρw) is shown above. Inside the air is pressurizedabove the water to a pressure Pt = 2Pa (where Pa is the air pressure outsideof the tank, one atmosphere). The water escapes through a small pipe atthe bottom where the stream emerges parallel to the ground as shown. Thecross-sectional area of the tank A is much larger than the cross-sectional area aof the small pipe at the bottom. Neglect viscosity and flow resistance. Pictureis not necessarily to scale.

a) Find the (approximate) speed va with which the water exits the smallpipe. You may assume A ≫ a.

b) Find the horizontal range of the stream of water, R, measured from thetip of the spout as shown. Express your answer in terms of va, so that itneedn’t depend on getting a) correct.


Problem 334.

problems/fluids-pr-bernoulli-sipping-through-straw.tex

40 cm

A

a

10 cm

When you drink through a straw, you create a pressure Pm in your mouth thatis less than atmospheric pressure. Suppose Pm = 9 × 104 Pa, and the end ofthe straw in your mouth is 10cm above the surface of your 40cm high drink asshown above. You may assume that the cross-sectional area of the straw a ismuch less than the cross-sectional area A of the fluid at the top of your glass.

a) At what speed will the fluid in the straw be moving into your mouth? (UseP0 = 105 Pa for the pressure of the air, ρ = 1000 kg/m3, g = 10 m/s2 andcompute a number after showing how you obtained an algebraicexpression for the answer.)

b) Find an algebraic expression for how long it will take to sip a small volume∆V of your drink through the straw. Assume that the fluid height in thecontainer makes a negligible change during this sip.

11.1. FLUIDS 385

Problem 335.

problems/fluids-pr-bernoulli-syringe.tex

F

P0

L

A

a

Pv

You are given a hypodermic syringe full of medicine that is basically a zero-viscosity fluid that is mostly water and has the same density. The syringe haslength L and cross-sectional area A and contains volume V = AL of fluid.Holding it horizontally in the room as shown, you press on the (frictionless)plunger to inject it into a patient’s vein where the blood pressure is Pv. Thecross-sectional area of the needle aperture is a ≪ A.

a) What force F (magnitude) do you have to exert on the plunger to holdthe fluid in static equilibrium once the needle is in the patient?

b) Suppose you exert six times that force, 6F . With what speed vp doesthe fluid flow into the patient? (Make any reasonable approximations tosimplify the algebra and/or arithmetic).

c) If one begins the injection at t = 0, at what time te will the syringe beempty?

d) After answering the questions above algebraically, for three points ofextra credit, evaluate F , vp, te, using A = 10−4 m2, a = 10−7 m2, P2 = 1.1atm, L = 0.05 m. You should not need a calculator.


Problem 336.

problems/fluids-pr-compare-barometers.tex

H

Pa

P = 0

The idea of a barometer is a simple one. A tube filled with a suitable liquid isinverted into a reservoir. The tube empties (maintaining a seal so air bubblescannot get into the tube) until the static pressure in the liquid is in balancewith the vacuum that forms at the top of the tube and the ambient pressure ofthe surrounding air on the fluid surface of the reservoir at the bottom.

a) Suppose the fluid is water, with ρw = 1000 kg/m3. Approximately howhigh will the water column be? Note that water is not an ideal fluid tomake a barometer with because of the height of the column necessaryand because of its annoying tendency to boil at room temperature into avacuum.

b) Suppose the fluid is mercury, with a specific gravity of 13.6. How highwill the mercury column be? Mercury, as you can see, is nearly idealfor fluids-pr-compare-barometers except for the minor problem with itsextreme toxicity and high vapor pressure.

Fortunately, there are many other ways of making good fluids-pr-compare-barometers.

11.1. FLUIDS 387

Problem 337.

problems/fluids-pr-crane-2.tex

m

A

M

A barge with a crane mounted on it has a cross sectional area A, a total mass M ,and straight sides. It is very slowly winching up a one of Blackbeard’s treasurechests (of total mass m) from the ocean floor near Beaufort.

a) As the chest comes out of the water, does the boat sink or rise? Justifyyour answer with an equation or two and/ or a before and after figure.

b) Just before the crane turns to put the chest on the deck, Blackbeard’sGhost appears and cuts the cable of the crane so that the chest plungesback into the briny deep. Find an expression for the distance d the boatrises up in the water (after it stops bobbing) when this happens. Use thesymbol ρs for the density of sea water.


Problem 338.

problems/fluids-pr-dangerous-drain.tex

0.5 m

r = 5 cm

10 m


• Pressure in a Static Fluid


That it is dangerous to build a drain for a pool or tub consisting of a singlenarrow pipe that drops down a long ways before encountering air at atmosphericpressure was demonstrated tragically in an accident that occurred (no fooling!)within two miles from where you are sitting (a baby pool was built with justsuch a drain, and was being drained one day when a little girl sat down on thedrain and was severely injured).

In this problem you will analyze why.

Suppose the mouth of a drain is a circle five centimeters in radius, and the poolhas been draining long enough that its drain pipe is filled with water (and nobubbles) to a depth of ten meters below the top of the drain, where it exits ina sewer line open to atmospheric pressure. The pool is 50 cm deep. If a thinsteel plate is dropped to suddenly cover the drain with a watertight seal, whatis the force one would have to exert to remove it straight up?

Note carefully this force relative to the likely strength of mere flesh and bone(or even thin steel plates!) Ignorance of physics can be actively dangerous.

11.1. FLUIDS 389

Problem 339.

problems/fluids-pr-firefighters-pump.tex

10 m

fire

pondpump truck

∆P

hose

Firefighters arrive at a fire in the country and have to use water from the farmpond to try to battle the blaze. Their pump firetruck takes in water from thepond at one atmosphere (P0) and increases the pressure at the bottom of thehose to an adjustable pressure P0 + ∆P that can be set at any value of ∆Pfrom 0 to 2 atmospheres of pressure. What is the minimum value ∆Pmin onecan set the pump to that will lift the water as high as the second floor (tenmeters up above the ground, two meters above the fire)? Show all work andjustify your answer with a physical principle or two!


Problem 340.

problems/fluids-pr-floating-freighter.tex

fresh saltd

A

A rectangular ocean barge with horizontal area A (viewed from the top) floatsin fresh water (ρw). It floats downriver and enters the ocean (ρs = 1.1ρw). As itdoes so, the ship bobs up an additional distance d from its earlier (freshwater)waterline. Find the total mass of the ship in terms of A, ρw, ρs and d. Hint –since you don’t know either the height of the ship or its displacement in freshwater as given, concentrate on the difference in the forces (and the displacement)as it sails from fresh to salt.

11.1. FLUIDS 391

Problem 341.

problems/fluids-pr-helium-balloon.tex

He

Air

In the figure above, a helium balloon (ρHe = 0.18 kg/m3) is suspended in air(ρa = 1.28 kg/m3) by a string.

a) Assuming that the volume of the helium balloon is approximately 4000cubic centimeters (4 × 10−3 m3), find the total ‘lift’ of the balloon (thetension in the string). Neglect the mass of the balloon itself and the string.

b) In the movies, humans are shown grabbing a few dozen helium balloonsand being pulled up into the sky. Assuming that a reasonable humanpayload (including the mass of all of the balloon rubber and strings) is100 kg, approximately how many balloons would really be required to lifta person?


Problem 342.

problems/fluids-pr-hot-air-balloon.tex

Cool Air

Hot Air

A hot air balloon is drawn in the figure above. Estimate its total ‘lift’, assumingthat the density of cool air is approximately constant at ρa = 1.28 kg/m3, thedensity of hot air in the baloon is ρh = 0.64 kg/m3, and that the balloon properhas a (filled) volume of 1000 m3 (corresponding to a spherical balloon roughly13 meters in diameter). If the balloon, basket, and rigging have a mass of 340kg, what is the maximum payload it can carry?

11.1. FLUIDS 393

Problem 343.

problems/fluids-pr-hydraulic-lift.tex

m

M

The figure above illustrates the principle of hydraulic lift. A pair of coupledcylinders are filled with an incompressible, very light fluid (assume that themass of the fluid is zero compared to everything else).

a) If the mass M on the left is 1000 kilograms, the cross-sectional area of theleft piston is 100 cm2, and the cross sectional area of the right piston is 1cm2, what mass m should one place on the right for the two objects to bein balance?

b) Suppose one pushes the right piston down a distance of one meter. Howmuch does the mass M rise?


Problem 344.

problems/fluids-pr-piston-pump-1.tex

M

v

A piston and weight has a total mass M and is pressing on water confined in acylinder of cross sectional area A = 100 cm2. The water is then pushed into apipe with a cross sectional area of a = 1 cm2 that is open to the air at the sameheight as the piston. Remember that ρw = 103 kg/meter3.

What does M have to be to make the water spurt from the pipe with a speed of5 meters/sec? Solve this problem beginning from (stated) physical principles,showing all work. You may use the approximation a ≪ A if it makes anythingeasier for you, and you obviously should solve the problem algebraically (andprobably finish the rest of the exam) before substituting any numbers.

11.1. FLUIDS 395

Problem 345.

problems/fluids-pr-piston-pump-2.tex

A

a

v

H

F

P0

A piston is pressed with a force ~F on a hydraulic cylinder containing water(ρ = 103 kg/m3). The cross sectional area of the cylinder is A = 400 cm2. Thewater therein is forced into a pipe with a cross sectional area of a = 2 cm2 thatrises vertically a height H = 40 meters. Both the end of the pipe (at the top)and the back of the piston (at the bottom) are open to atmospheric pressure.

What does F have to be to make the water spurt from the pipe with a speed of10 meters/sec at the top? Solve this problem beginning from (stated) physicalprinciples, showing all work.


Problem 346.

problems/fluids-pr-pump-water-up-cliff.tex

H

Pump?Outflow

pool h = 8 m


• Static Pressure

• Barometers


A pump is a machine that can maintain a pressure differential between its twosides. A particular pump that can maintain a pressure differential of as much as10 atmospheres of pressure between the low pressure side and the high pressureside is being used on a construction site.

a) Your construction boss has just called you into her office to either explainwhy they aren’t getting any water out of the pump on top of the H = 25 meterhigh cliff shown above. Examine the schematic above and show (algebraically)why it cannot possibly deliver water that high. Your explanation should includean invocation of the appropriate physical law(s) and an explicit calculation ofthe highest distance the a pump could lift water in this arrangement. Why isthe notion that the pump “sucks water up” misleading? What really moves thewater up?

b) If you answered a), you get to keep your job. If you answer b), you mighteven get a raise (or at least, get full credit on this problem)! Tell your bosswhere this single pump should be located to move water up to the top and show(draw a picture of) how it should be hooked up.

11.1. FLUIDS 397

Problem 347.

problems/fluids-pr-romeo-and-juliet.tex

DH

hole

Romeo and Juliet are out in their boat again when Juliet’s Salvatore Ferragamoheels poke a circular hole of radius r in the bottom of the boat. The boat has adraft of D (this is the distance the boat’s bottom lies underwater as shown).

a) Romeo tries to cover the hole with his hand. What is the minimum forcehe must apply to keep it covered?

b) Juliet convinces Romeo that a little water fountain would be romantic, sohe moves his hand. How fast does the water move through the hole?

c) To what height H does Juliet’s fountain spout up from the bottom of theboat? (The height drawn is to illustrate the quantity H only and may notbe at all correct.)


Problem 348.

problems/fluids-pr-siphon.tex

d

h

Water is being drained from a large container by means of a siphon as shown.The highest point in the siphon is distance d above the level of water in thecontainer, and the total height of the long arm of the siphon is h. The distanceh can be varied. The mass density of water is ρw, and air pressure is P0. Expressall answers in terms of d, h, ρw, P0, and g.

a) What is the maximum possible value of h for which the siphon will work?(Hint: The pressure cannot be negative anywhere in the siphon, in par-ticular, in the long arm of the siphon.)

b) For that maximum value of h, what is the speed of the water coming outof the siphon?

11.1. FLUIDS 399

Problem 349.

problems/fluids-pr-static-crane.tex

m

D

d

M

The crane above has a nearly massless boom. It is being used to salvage someof Blackbeard’s treasure – a chest of mass m filled with very dense gold.

a) Find the maximum weight that the crane can lift, assuming that all of theweight of the crane itself acts downward at its center of mass to counter-balance it at the position shown, a horizontal distance d to the left of thebottom right corner of the crane. The crane’s boom is fixed so that itsmoment arm (shown) is always D. Your answer should be expressed inM , g and the given lengths d and D.

b) Suppose that Blackbeard’s treasure is so massive that the crane is almost

tipping over as it very slowly lifts it up through the water. What willhappen when the crane tries to lift the mass out of the water, and why?“Why” should involve certain forces and a good before and after picture.


Problem 350.

problems/fluids-pr-static-hole-in-a-boat.tex

2m

Fresh water

patch (10 cm )

screws, bracing

Air

Boat

Your yacht has a hole in it! Oh, no! The hole is 2 meters below the waterline,and has a cross-sectional area of 10 cm2 (that’s ten square centimeters, not tencentimeter’s squared!). You patch it, and need to brace the patch with screwsthat can each hold at most a force of 5 Newtons. How many screws (at least)should you use to be sure of being able to withstand the force of the oceanpressing in against your patch?

11.1. FLUIDS 401

Problem 351.

problems/fluids-pr-time-to-empty-open-vat.tex

0

0H

AP

a

P


• Bernoulli’s Equation

• Torricelli’s Law


In the figure above, a large drum of water is open at the top and filled upto a height H above a tap at the bottom (which is also open to normal airpressure). The drum has a cross-sectional area A at the top and the tap has across sectional area of a at the bottom.

a) Find the speed with which the water emerges from the tap. Assumelaminar flow without resistance. Compare your answer to the speed amass has after falling a height H in a uniform gravitational field (afterusing A ≫ a to simplify your final answer, Torricelli’s Law).

b) How long does it take for all of the water to flow out of the tap? (Hint:Start by guessing a reasonable answer using dimensional analysis andinsight gained from a). That is, think about how you expect the time tovary with each quantity and form a simple expression with the relevantparameters that has the right units. Next, find an expression for thevelocity of the top. Integrate to find the time it takes for the top to reachthe bottom.) Compare your answer(s) to each other and the time it takesa mass to fall a height H in a uniform gravitational field. Does the correctanswer make dimensional and physical sense?

c) Evaluate the answers to a) and b) for A = 0.50 m2, a = 0.5 cm2, H = 100cm.


Problem 352.

problems/fluids-pr-weight-of-immersed-mass.tex

W?

T?


• Archimedes Principle

• Weight


A block of density ρ and volume V is suspended by a thin thread and is immersedcompletely in a jar of oil (density ρo < ρ) that is resting on a scale as shown.The total mass of the oil and jar (alone) is M .

a) What is the buoyant force exerted by the oil on the block?

b) What is the tension T in the thread?

c) What does the scale read?

Chapter 12

Oscillations

And now for one of the most important physical topics ever. We have seen thatstatics is pretty important. Atoms bond together to make molecules or solidsthat are in a sort of static equilibrium. Objects are glued, or stapled, or nailedtogether into bigger objects in static equilibrium. Anything that has persistentstable structure lives in, or near, a state of static equilibrium.

Did I just say near? I did. If you pull any system a little bit out of a stableequilibrium, it will usually be pushed back towards its equilibrium positionbecause it is stable. Furthermore, our friend the Taylor Series tells us thatmost often the restoring force (or torque) will be linear in the displacementfor sufficiently small displacements from equilibrium. And what happens whenyou displace any mass from a stable equilibrium so that it experiences a linearrestoring force?

It oscillates. Harmonically.

That doesn’t mean it plays the harmonica – it means that the oscillation canbe described by the harmonic functions”, sine, cosine, and the (complex) expo-nential.

Oooo, I just used a bad word – complex exponential. Sorry, but in order tomaster a variety of concepts associated with oscillation and (very soon) waves,it really helps if you know something about complex numbers. And, of course,harmonic functions. And some trig identities that you almost certainly haveforgotten. Time to go brush up on all of this stuff, or (if necessary) learn it forthe first time.

403

404 CHAPTER 12. OSCILLATIONS

12.1 Oscillations


Problem 353.

problems/oscillation-mc-change-resonance-3.tex

P

ωω0

∆ω

avg

In the figure above, the curve shows the (average) power Pavg(ω) delivered to adamped, driven oscillator with equation of motion:

md2x

dt2+ b

dx

dt+ kx = F0 cos(ωt)

Recall that the “width” of the curve ∆ω is the full width at half maximumpower. Suppose the damping constant b is doubled while k of the spring, m,and the driving force magnitude F0 are kept unchanged. What happens tothe curve?

a) The curve becomes narrower (smaller ∆ω) at the same frequency;

b) The curve becomes narrower at a higher frequency;

c) The curve becomes broader (larger ∆ω) at the same frequency

d) The curve becomes broader at a different frequency;

e) The curve does not change;

f) There is not enough information to determine the changes of the curve.

12.1. OSCILLATIONS 405

Problem 354.

problems/oscillation-mc-increase-damping-resonance-curve.tex

P

ωω0

∆ω

avg

In the figure above, the curve shows the (average) power Pavg(ω) delivered to adriven damped oscillator. Recall that the “width” of the curve ∆ω is the fullwidth at half maximum. If damping is increased while everything else iskept unchanged, what happens to the curve?

a) The curve becomes taller and wider;

b) The curve becomes taller and narrower;

c) The curve becomes shorter and wider;

d) The curve becomes shorter and narrower;




Problem 355.

problems/oscillation-mc-increase-k-m-resonance-curve.tex

P

ωω0

∆ω

avg

In the figure above, the curve shows the (average) power Pavg(ω) delivered to adamped, driven oscillator with equation of motion:

md2x

dt2+ b

dx

dt+ kx = F0 cos(ωt)

Recall that the “width” of the curve ∆ω is the full width at half maximumpower. If both k of the spring and m are doubled while the damping constantb and driving force magnitude F0 are kept unchanged, what happens to thecurve?

a) The curve becomes narrower (smaller ∆ω) at the same frequency;

b) The curve becomes narrower at a higher frequency;

c) The curve becomes broader (larger ∆ω) at the same frequency

d) The curve becomes broader at a different frequency;




Problem 356.

problems/oscillation-mc-stride-resonance.tex

You have to take a long hike on level ground, and are in a hurry to finish it. Onthe other hand, you don’t want to waste energy and arrive more tired than youhave to be.

Your stride is the length of your steps. Your pace is the frequency of your steps,basically the number of steps you take per minute. Your average speed is theproduct of your pace and your stride: the distance travelled per minute is thenumber of steps you take per minute times the distance you cover per step.

Your best strategy to cover the distance faster but with minimum additionalenergy consumed is to:

a) Increase your stride but keep your pace about the same.

b) Increase your pace, but keep your stride about the same.

c) Increase your pace and your stride.

d) Increase your stride but decrease your pace.

e) Increase your pace but decrease your stride.

(in all cases so that your average speed increases).

Note well that this is a physics problem, so be sure to justify your answer witha physical argument. You might want to think about why one answer willprobably accomplish your goal within the constraints and the others will not.


Problem 357.

problems/oscillation-mc-two-damped-oscillators.tex

m 2m

same

A B

Two identical springs support two masses of the same size and shape inthe same damping fluid. However, mB = 2mA.

Both systems are pulled to an initial displacement from equilibrium of X0 andreleased, and the exponential decay times τA and τB required for the initialamplitude of oscillation of each mass to decay to X0e

−1 is measured. We expectthat:

a) τA = 2τB

b) 2τA = τB

c) τA = τB

d) 4τA = τB

e) We cannot predict the relative decay times without more information.



Problem 358.

problems/oscillation-ra-compressed-rods-youngs-modulus-scaling.tex

r r 2r 2r

L

2L

L/2L

A B C D

F

F

In the figure above rank the compression ∆L of the rods shown when an identicalforce with magnitude F is exerted between the ends (as shown in D). Equalityis a possibility. Your answer should look something like C = D > A > B.


Problem 359.

problems/oscillation-ra-compression-three-rods-1.tex

m

r

L

L/2

2L

2r

r/2

cba

In the figure above three rods made out of copper are shown with the dimensionsgiven. In (a), a mass m is placed on top of the rod (which rests on a rigid table)and the rod is observed to be compressed and shrinks by a length ∆L. By whatlength do you expect rods (b) and (c) to be compressed by if the same mass mis placed on top of them? (Express your answer in terms of ∆L.)

b:

c:


Problem 360.

problems/oscillation-ra-mass-spring-double-displacement.tex

Two identical masses are attached to two identical springs. The first mass ispulled to a distance x0 from equilibrium. The second one is pulled to a distance2x0 from equilibrium. At time t = 0 they are released. The first mass reachesits equilibrium point at time t1, the second one at time t2.

What is the ratio t2/t1?


Problem 361.

problems/oscillation-ra-mass-swing-double-displacement.tex

Two kids are sitting on swings of equal length. One of them has about twicethe mass of the other (but they are about the same height). The lighter one ispulled back to an initial (small) angle θ0. The heavier one is pulled back to a(still small!) angle 2θ0. At t = 0 they are both released. It takes the lighter onea time tl to reach the lowest point of his trajectory, and the heavier one a timeth.

What is the ratio th/tl?


Problem 362.

problems/oscillation-ra-physical-pendula-periods.tex

A B C D

In the figures above, four physical pendulums are drawn. All consist of a light(massless) rod of length L to the center of mass of different shaped massesconnected to the end. All of the shapes have the same mass M and the sameprimary length scale R. Rank the periods of the physical pendulums fromlowest (highest frequency!) to the highest (lowest frequency!). Equality is apossibility.

The moments of inertia of the round objects (about their centers of mass) are:

A) I = 12MR2 (disk)

B) I = MR2 (hoop)

C) I = 23MR2 (hollow ball)

D) I = 25MR2 (solid ball)


Problem 363.

problems/oscillation-ra-physical-pendulums-1.tex

L/2

L/4

L

M

a b c

In the figure above, three pendulums are suspended from frictionless pivots. Thefirst is a rod of mass M and length L. The second is a “point” mass M withnegligible radius. The third is a disk of mass M and radius L/2. In all threecases, the center of mass of the pendulum is a distance L/2 from the pivot andthe mass is constrained to rotate around the pivot (physical pendulum). Rankthe angular frequencies (where equality is allowed) so that an answer might be(but probably isn’t) ωa > ωb = ωc.


Problem 364.

problems/oscillation-ra-physical-pendulums-2.tex

M

a b c

L

L/4

In the figure above, three pendulums are suspended from frictionless pivots.The first is a thick rod of mass M and length L. The second is a “point” massM with negligible radius on a thin (massless) rod of length L. The third is adisk of mass M and radius L/4 on the end of a thin (massless) rod of so thatits center of mass is a distance L away from the pivot. In all three cases, themass is constrained to rotate around the pivot as a physical pendulum.

Rank the angular frequencies in increasing order (where equality is al-lowed) so that an answer might be (but probably isn’t) ωa > ωb = ωc.


Problem 365.

problems/oscillation-ra-rank-the-damped-periods.tex

CA B(air)(vacuum) (water)

In the figure above identical masses are connected to identical springs and lo-cated in three different labelled containers. All three masses are pulled to thesame distance from equilibrium and are released from rest. The container Acontains a vacuum, container B is filled with ordinary room-temperature air at1 atmosphere of pressure, and container C contains water.

Rank the period of the oscillation of the three masses by their container number,where (precise) equality is a possibility. That is, a possible answer might beTa = Tc < Tb (but probably isn’t). Explain your answer with a few words or anequation.


Problem 366.

problems/oscillation-ra-rank-the-periods.tex

m

2m

kk

CD

m mk

kk k

A B

g

In the figure above, rank the periods of each pair of oscillators shown (whereequality is allowed). That is, fill in the boxes in the two expressions below witha <, >, = sign as appropriate.

TA TB TC TD


Problem 367.

problems/oscillation-ra-series-parallel-frequency-easy.tex

m

m

mk k

k k kA

B

k

k

k

C k

k

m

D

Rank the oscillation frequencies of the identical masses m connected to thesprings in the figure above from lowest to highest with equality a possibility.The springs have spring constant k, and you should neglect damping. A possibleanswer is (as always) D < A = B < C or the like.


Problem 368.

problems/oscillation-ra-series-parallel-frequency.tex

k k k

k

k

k

k k

k

k

B

C

A

D

m m/2

2m3m

Rank the frequencies of the masses on the spring arrangements in the figureabove, from lowest to highest with equality a possibility. Neglect damping.A possible answer is (as always) D < A = B < C or the like.


Problem 369.

problems/oscillation-ra-series-parallel-period.tex

k k k

k

k

k

k k

k

k

B

C

A

D

m m/2

2m3m

Rank the period of oscillation of the masses on the spring arrangements inthe figure above, from lowest to highest with equality a possibility. Neglectdamping. A possible answer could be (as always) D < A = B < C but probablyisn’t.


Problem 370.

problems/oscillation-ra-shear-three-rods-1.tex

Lh

w

2h

w w/2

h/2 L/22L

FF F

a b c

In the figure above, three light wooden boards and their relative dimensions areshown. The boards are each fixed in a vise (not shown) on the left hand side

so that the left end of each board cannot move. A downward force ~F is appliedat the right hand end of each board. The first board is bent by this force sothat its right hand end is displaced downward by a distance ∆h. By how muchare the right hand ends of the other two boards displaced downward? (Expressyour answer in terms of ∆h.)

b:

c:


12.1.3 Short Answer

Problem 371.

problems/oscillation-sa-damping-variation.tex

k

bm

The damped oscillator above is set in motion at time t = 0. Fill in the followingtable with x’s in the provided boxes. τ is the exponential damping time of theamplitude, and ω0 is the natural frequency.:

If b increases: τ increases decreases remains unchanged. ω0 increases de-creases remains unchanged

If m increases: τ increases decreases remains unchanged. ω0 increases de-creases remains unchanged

If k increases: τ increases decreases remains unchanged. ω0 increases de-creases remains unchanged


Problem 372.

problems/oscillation-sa-estimate-Q-resonance-curve.tex

1

2

3

4

5

6

7

8

9

10

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

P(

ω)

ω

(a)

(b)

(c)

In the figure above, three resonance curves showing the amplitude of steady-state driven oscillation A(ω) as functions of ω. In all three cases the resonancefrequency ω0 is the same. Put down an estimate of the Q-value of each oscillatorby looking at the graph. It may help for you to put down the definition of Qmost relevant to the process of estimation on the page.

a)

b)

c)


Problem 373.

problems/oscillation-sa-match-the-waveform.tex

You are presented with three identical simple harmonic oscillators, A,B,C,which oscillate with a known harmonic frequency ω. They differ only in theirinitial conditions. At time t = 0, the attached masses have an initial positionand velocity (x, v) given by:

A (xA = x0vA = 0)

B (xB = 0, vB = v0)

C (xC = x0, vC = x0 ∗ ω)

where x0 and v0 are positive numbers not equal to zero in the appropriateunits.

Match each set of initial conditions to the corresponding solution from the listof possible solution forms (put A, B or C in three of the four boxes) below:

x(t) = A cos(ωt)

x(t) = A cos(ωt − π/4)

x(t) = A sin(ωt)

x(t) = A cos(ωt + π/4)

Consider one of oscillator solutions, x(t) = Asin(ωt). Find the velocity andacceleration at t = T/4, where T is the period of oscillations: that is, pleaseevaluate vx(t = T/4) and ax(t = T/4).


Problem 374.

problems/oscillation-sa-roman-soldiers-bridge-resonance.tex

Roman soldiers (like soldiers the world over even today) marched in step at aconstant frequency – except when crossing wooden bridges, when they broketheir march and walked over with random pacing. Why? What might havehappened (and originally did sometimes happen) if they marched across with acollective periodic step?


Problem 375.

problems/oscillation-sa-series-parallel-frequency.tex

k k

k

k

k

B

m

3m

m/2

D

3k

A

k

k

C

2m

Find the ratio of the angular frequencies of each spring-mass combination aboveto ω0 =

√

k/m.

ωAω0

=

ωBω0

=

ωCω0

=

ωDω0

=


Problem 376.

problems/oscillation-sa-sho-true-facts.tex

The one-dimensional motion of a mass m is described by x(t) = A sin(ωt).Identify the true and false statements among the following by placing a T inthe provided box for true statements and an F in the provided box for falsestatements:

a) If A and ω are constant (i.e. – independent of time t) the motion

is simple harmonic motion.

b) The mass m starts at t = 0 with zero velocity.

c) If the motion of mass m is simple harmonic oscillation, the poten-

tial energy of the mass can be written U(x) = 12mω2x2.

d) If the motion of the mass m is simple harmonic oscillation, the

total force acting on the mass can be written Fx = −mω2x.


Problem 377.

problems/oscillation-sa-sketch-damped-oscillation.tex

t

x(t)

t

x(t)

t

x(t)

k

bm

(b)

(a)

(c) (inset)

A mass m is attached to a spring with spring constant k and immersed in adamping fluid with linear damping coefficient b as shown in the inset figureabove. Equilibrium is at x = 0 meters. At time t = 0 seconds the mass ispulled to x(0) = 1 meters and released from rest. The period of the oscillatorin the absence of damping is T = 2 seconds. On the provided axes with integertick-marks above, sketch the following:

a) x(t) in the absence of damping.

b) x(t) if b/2m = 1/3 (underdamped, assume that ω′ ≈ ω0).

c) x(t) in the case where b/2m = π (critically damped).

The second two curves only need to be qualitatively correct (you don’t have toplot them exactly), but they should also not be crazily out of scale. You may usee = 2.72 ≈ 3 to make drawing the curves easier without needing a calculator.


Problem 378.

problems/oscillation-sa-sketch-oscillation.tex

t

x(t)

(b)

(a)

(c) (inset)

t

t

v(t)

a(t)

k

m

A mass m is attached to a spring with spring constant k as shown in the insetfigure above. There is no damping.

On the provided axes above, sketch x(t), v(t) and a(t), given that at time t = 0the mass is pulled to x(0) = X0 = 1 (relative to equilibrium) and released fromrest. The period is T = 2 seconds, and you should use the tic-marks on the taxis as seconds. Your graphs should have the correct sign, phase, period, andyou should label the peak positive value in terms of the givens on the ordinateaxes.



Problem 379.

problems/oscillation-pr-bar-and-spring-1.tex

θ

LM

k s

In the figure above a rigid rod of mass M and length L is pivoted in the centerwith a frictionless bearing. Its lower end is attached to a spring with springconstant k as shown that is unstretched (at equilibrium) when the rod is verticaland θ = 0.

For small displacements s ≪ L (where one can use the small angle approxima-tion), the spring will exert a restoring force Fs = −ks ≈ −k(L/2)θ along thearc of motion of the end of the rod. It is pulled to an initial small displacementangle θ0 and released at time t = 0.

a) What is the period of this oscillator for small oscillations?

b) What is the angular velocity Ω of the rod when it reaches its equilibriumposition at θ = 0? (Note well: Do not confuse ω0, the angular frequencyof oscillation, and Ω = dθ

dt , the angular velocity of the rod! Don’t forgetdirection!)


Problem 380.

problems/oscillation-pr-bar-and-spring-2.tex

k

L

θ

s

m

In the figure above a rigid rod of mass m and length L is pivoted at the end witha frictionless bearing. Its lower end is attached to a spring with spring constantk as shown that is unstretched (at equilibrium) when the rod is vertical andθ = 0.

For small displacements s ≪ L (where one can use the small angle approxima-tion), the spring will exert a restoring force Fs = −ks along the arc of motionof the end of the rod. It is pulled to an initial small displacement angle θ0 andreleased at time t = 0, at which point it will begin to oscillate with angularfrequency ω0.

a) Neglecting damping, find the period T0 of this oscillator for small os-cillations and sketch a qualitatively correct graph of θ(t) for the rod.(Note well: both the spring and gravity contribute to the motion of therod!)

b) What is the angular velocity of the rod ω = dθdt when it reaches its equi-

librium position at θ = 0? Do not confuse the angular velocity of the rodwith its angular frequency.

c) Suppose one compares the predicted motion θ(t) to the motion one wouldactually observe in the real world, where the system surely would be atleast weakly damped. Sketch a graph that is qualitatively correctillustrating what θ(t) might really look like when weak damping is takeninto account.


(Hint: The moment of inertia of a rod pivoted about one end is 13ML2.)


Problem 381.

problems/oscillation-pr-block-on-vertically-oscillating-plate.tex

Mm

k

A block of mass m is sitting on a plate of mass M . It is supported by a verticalideal massless spring with spring constant k. Gravity points down.

a) When the system is at rest, how much is the spring compressed from itscompletely uncompressed length?

b) The spring is pushed down an extra distance A and released. Assumingthat the mass m remains on the plate, what is its frequency of verticaloscillation?

c) What is the maximum value of A such that the small mass m will notleave the plate at any point in the motion?

Express all answers in terms of m, M, k, g.


Problem 382.

problems/oscillation-pr-box-of-springs.tex

m

k k

kk

?

You are given a mass m, a box full of identical springs each with spring constantk, and a bunch of stiff wire you can bend and use to fasten the springs togetherto the wall and the mass in any combination of series and parallel you like.

I’ve drawn one such arrangement for you, one that will cause the mass m tooscillate harmonically on a smooth surface at angular frequency ω. Your jobis to design an arrangement of springs that will make the mass oscillate at an

angular frequency of√

3k2m , using only the (uncut) springs in the box.

a) Find the angular frequency of the four-spring oscillator I’ve drawn.

b) Draw a new arrangement on the bar underneath (or elsewhere on your

paper) that will have an angular frequency of√

3k2m . Note well that there

is more than one way to get the right answer, but some ways need (a lot)more springs than others. Try to get an answer with no more than sixsprings

c) Prove/show that your answer is correct.


Problem 383.

problems/oscillation-pr-car-on-springs-resonance.tex

A car with a mass of M = 1000 kg rests on shock absorber springs with acollective spring constant of k = 105 N/m. It is driving down a road whichhas raised expansion joints every 5 meters that bounce the car. At what speedwould you expect the ride to be roughest?


Problem 384.

problems/oscillation-pr-damped-oscillation.tex

k

bm

A mass m is attached to a spring with spring constant k and immersed in amedium with damping coefficient b. (Gravity, if present at all, is irrelevant asshown in class). The net force on the mass when displaced by x from equilibriumand moving with velocity vx is thus:

Fx = max = −kx − bvx

(in one dimension).

a) Convert this equation (Newton’s second law for the mass/spring/dampingfluid arrangement) into the equation of motion for the system, a “secondorder linear homogeneous differential equation” as done in class.

b) Optionally solve this equation, finding in particular the exponential damp-ing rate of the solution (the real part of the exponential time constant)and the shifted frequency ω′, assuming that the motion is underdamped.You can put down any form you like for the answer; the easiest is probablya sum of exponential forms. However, you may also simply put downthe solution derived in class if you plan to just memorize this solutioninstead of learn to derive and understand it.

c) Using your answer for ω′ from part b), write down the criteria for damped,underdamped, and critically damped oscillation.

d) Draw three qualitatively correct graphs of x(t) if the oscillator is pulledto a position x0 and released at rest at time t = 0, one for each damping.Note that you should be able to do this part even if you cannot derive thecurves that you draw or ω′. Clearly label each curve.


Problem 385.

problems/oscillation-pr-disk-with-rim-weight.tex

R θP

m

2m

A uniform disk of radius R and mass 2m can freely rotate about a fixed fric-tionless horizontal axis passing through its fixed center P as shown. It has apoint mass m fixed on its rim, so that in equilibrium, the disk is oriented suchthat θ = 0. At time t = 0, the disk is gently rotated by a small angle θ0 andreleased from rest.

a) Just after it is released, what is the net torque ~τ about P acting on thedisk?

b) After the disk is released, it oscillates. What is the angular frequency ωof the oscillation?

c) Find θ(t), i.e., the angular position of the point mass as a function of time.


Problem 386.

problems/oscillation-pr-inelastic-collision-mass-on-spring-2.tex

k

M

x

vm

collision

block comes to rest

A block of mass M that is sitting on a frictionless table, at rest, attached to aspring of mass k at its equilibrium position as shown. A bullet of mass m andvelocity v (to the right) penetrates the block and “sticks” in a hole prepared tocatch it.

a) Find the velocity of the block and bullet as they move together immedi-ately after the collision (before they have started to compress the spring).

b) How much kinetic energy (if any) was lost in this collision?

c) Find the distance x that the spring will compress before the bullet/blockcome momentarily to rest.


Problem 387.

problems/oscillation-pr-inelastic-collision-mass-on-spring.tex

mv

collision

xx = 0

M−m

A bullet of mass m, travelling at speed v, hits a block of mass M−m with a pre-drilled hole resting connected at the equilibrium position to a connected springwith constant k and sticks in the hole. The block is sitting on a frictionlesstable (i.e. – ignore damping). Assume that the collision occurs at t = 0. Allanswers below should be given in terms of m, M, k, v.

a) What is the maximum displacement X0 of the block?

b) What is the angular frequency ω of oscillation of the combined bullet-blocksystem?

c) Write down x(t), the position of the block as a function of time.


Problem 388.

problems/oscillation-pr-mass-on-spring-damped.tex

k

bm

A mass m is attached to a spring with spring constant k and immersed in amedium with damping coefficient b. The net force on the mass when displacedby x from its equilibrium position is thus:

Fx = max = −kx − bvx

Convert this equation (Newton’s second law for the mass/spring/damping fluidarrangement) into a second order linear homogeneous differential equation andsolve it, finding the damping rate and the shifted frequency ω′. You may leavethe final answer in exponential form or convert it to cosine as you wish.

Also Draw a qualitatively correct graph of x(t) if the oscillator is pulled to aposition x0 and released at rest at time t = 0. Note that you should be able todo this part even if you cannot derive the curves that you draw or ω′.


Problem 389.

problems/oscillation-pr-minimize-period-of-disk.tex

M

R

pivot

x θ

A uniform disk of mass M and radius R has a hole drilled in it a distance0 ≤ x < R from its center. It is then hung on a (frictionless) pivot, pulled to theside through a small angle θ0, and released from rest to oscillate harmonically.

a) What is the moment of inertia of the disk about this pivot?

b) Write τ = Iα for this disk, make the small angle approximation, and turnit into the differential equation of motion.

c) Write an expression for T , the period of oscillation of the disk, as a functionof d.

d) 5 point extra credit bonus question! What value of d minimizes thisperiod? That is, if we wanted to make a disk oscillate with the shortestpossible period, how far from the end would we drill a pivot hole?


Problem 390.

problems/oscillation-pr-minimize-period-of-rod.tex

L

M

d

θ

pivot

CM

A rod of mass M and length L is pivoted a distance d from the center as shownabove. Gravity acts on the rod, pulling it down (as usual) at its center of mass.

a) What is the moment of inertia of the rod about this pivot?

b) Write τ = Iα for this rod, make the small angle approximation, and turnit into the differential equation of motion.

c) Write an expression for T , the period of oscillation of the rod, as a functionof d.

d) 5 point extra credit bonus question! What value of d minimizes thisperiod? That is, if we wanted to make a rod oscillate with the shortestpossible period, how far from the end would we drill a pivot hole?


Problem 391.

problems/oscillation-pr-pendulum-with-spring.tex

sk

m

θ

L

In the figure above a mass m on the end of a massless string of length L formsa pendulum. A light (massless) spring of spring constant k is attached to themass so that for small oscillations s ≪ L (where one can use the small angleapproximation), Fs = −ks where s is the distance along the arc of motion fromthe equilibrium position in the center. When released, both gravity and thespring contribute to its motion, with the force exerted by the spring remainingapproximately tangent to the trajectory throughout.

a) Find the period of this oscillator for small oscillations.

b) If it is started at an angle θ0 and released, how fast is the mass m movingas it crosses equilibrium at θ = 0?


Problem 392.

problems/oscillation-pr-physical-pendulum-ball-on-stick.tex

θ0

L

R

M

A physical pendulum is constructed from a thin rod of negligible mass insertedinto a uniform ball of mass M and radius R. The rod has length L from thepivot point to the center of the ball. At time t = 0 the ball is released fromrest when the rod is at an initial small angle θ0 with respect to its verticalequilibrium position.

Answer all the questions below in terms of M, R, L, g, θ0. You may make thesmall angle approximation where appropriate.

a) Determine the equation of motion for the system, solving for α = d2θdt2 .

b) Determine the angular frequency of oscillation ω and write down θ(t) forthe ball.

c) Find the maximum speed v of the ball. Is this larger or smaller than itwould have been if the ball had been a point mass M at the end of therod? Why?


Problem 393.

problems/oscillation-pr-physical-pendulum-disk-on-stick-1.tex

R

M

L

θ0

A physical pendulum is constructed from a thin rod of negligible mass rigidlyinserted into a uniform disk of mass M and radius R. The rod has length Lfrom the pivot point at the top of the rod to the center of the disk. At timet = 0 the disk is released from rest when the rod is at an initial small angle θ0

with respect to its vertical equilibrium position.

Answer all the questions below in terms of M, R, L, g, θ0. You may make thesmall angle approximation where appropriate.

a) Find the vector torque ~τ about the pivot point at the instant the ball is

released, assuming θ0 > 0 (positive) as drawn.

b) Determine the period T of the resulting oscillation.

c) Find the maximum speed v of the center of mass of the disk as itoscillates.


Problem 394.

problems/oscillation-pr-physical-pendulum-disk-on-stick.tex

R

M

L

θ0

A physical pendulum is constructed from a thin rod of negligible mass insertedinto a uniform disk of mass M and radius R. The rod has length L from the pivotpoint to the center of the disk. At time t = 0 the disk is released from rest whenthe rod is at an initial small angle θ0 with respect to its vertical equilibriumposition. You may make the small angle approximation where appropriate.

a) Determine the equation of motion for the system, solving for α = d2θdt2 .

b) Determine the angular frequency of oscillation and write down the har-monic motion solution θ(t) for the disk.

c) Find the maximum speed v of the disk.

d) Is this larger or smaller than it would have been if the ball had been apoint mass M at the end of the rod? Why?


Problem 395.

problems/oscillation-pr-physical-pendulum-disk-on-stick-v0-only.tex

M

L

R v0

A physical pendulum is constructed from a thin rod of negligible mass insertedinto a uniform disk of mass M and radius R. The rod has length L from thepivot point to the center of the disk. At time t = 0 the disk is sitting in itsequilibrium position θ = 0 and is given as sharp blow so that it has an initialspeed of v0 to the right. The resulting oscillation is “small”; you may makethe small angle approximation where appropriate.

a) Draw the situation at a time that the pendulum has swung through anarbitrary angle θ. Determine the equation of motion for the system,

solving for α = d2θdt2 .

b) Determine the angular frequency of oscillation and write down the har-monic motion solution θ(t) for the disk. (Hint: What is the maximumangular velocity of the pendulum?)

c) Find the maximum angle θmax that the disk reaches.

d) Is angle θmax larger or smaller than it would have been if the ball hadbeen a point mass M at the end of the rod started with the same initialvelocity? Why?


Problem 396.

problems/oscillation-pr-physical-pendulum-disk-with-hole.tex

R

pivot

x

θ

M

0

A uniform disk of mass M and radius R has a hole drilled in it a distance0 ≤ x < R from its center. It is then hung on a (frictionless) pivot, pulled to theside through a small angle θ0, and released from rest to oscillate harmonically.

a) What is the moment of inertia of the disk around this pivot?

b) Write down the differential equation of motion for this physical pendulum.Circle ω2.

c) Find the period of the physical pendulum as a function of (possibly) x,M , R, and g.

d) Write down the solution to the equation of motion, θ(t).


Problem 397.

problems/oscillation-pr-physical-pendulum-grandfather-clock.tex

θ0

L

R

M

A Grandfather clock’s pendulum is constructed from a thin rod of negligiblemass inserted into a uniform disk of mass M = 1 kg and radius R = 5 cm. Therod has a length L from the pivot point to the center of the disk that can beadjusted from 0.20 m to 0.30 m in length so that the clock keeps the correcttime.

a) Algebraically determine the (differential) equation of motion for the sys-tem, making the small angle approximation to put it in the form of a simpleharmonic oscillator equation.

b) The clock keeps correct time when the period of its pendulum is T = 1second. What should L be (to 3 digits) so that this is true. (Use thealgebraic form for ω2 from your answer to part a to solve for L.)


Problem 398.

problems/oscillation-pr-rolling-wheel-1.tex

kM

y

Rx

A spring with spring constant k is attached to a wall and to the axle of a wheelof radius R, mass M , and moment of inertia I = βMR2 that is sitting on arough floor. The wheel is stretched a distance A from its equilibrium positionand is released at rest at time t = 0. The rough floor provides enough staticfriction that the wheel rolls without slipping.

a) When the displacement of the wheel from its equilibrium position is x andthe speed of center of mass of the wheel is v, what is its total mechanicalenergy?

b) What is the maximum velocity vmax of the wheel?

c) What is the angular frequency of oscillation, ω, for the axle of the wheelas it rolls back and forth? (Note that this is not the angular velocity ofthe rolling wheel!)


Problem 399.

problems/oscillation-pr-rolling-wheel-and-spring.tex

kM

R

x0

A spring with spring constant k is attached to a wall and to the axle of a wheelof radius R, mass M , and moment of inertia I = βMR2 that is sitting on arough floor. The wheel is stretched a distance A from its equilibrium positionand is released at rest at time t = 0. The rough floor provides enough staticfriction that the wheel rolls without slipping.

a) What is the angular frequency of oscillation, ω, for the wheel as it rollsback and forth? (Note that this is not the angular velocity of the rollingwheel!)

b) What is the total energy of the wheel?

c) What is the maximum velocity vmax of the wheel?


Problem 400.

problems/oscillation-pr-three-block-inelastic-collision.tex

+x

L/2 L/2

v0

k

m m 2m

Two blocks of mass m and 2m are resting on a frictionless table, connected byan ideal (massless) spring with spring constant k at its equilibrium length L. Athird block of mass m is moving to the right with speed v0 as shown. It collideswith and sticks to the block of mass m connected to the spring (forming a new“block” of mass 2m on the left hand end of the spring).

We wish to find the position of both the left and the right hand blocks asfunctions of time. This is a challenging problem and will require several stepsof work. Hints: Think about what is conserved both during the collisionand during the subsequent motion of the blocks. Try to visualize this motion.Finally, the motion of the blocks is simplest in the center of mass frame.

The following questions will guide you through the work:

a) Let the origin of the laboratory frame be the location of the center of massof the system at the instant of collision. Write an expression for xcm(t),the position of the center of mass as a function of time.

b) What is the total kinetic energy of the system immediately after thecollision?

c) What is the kinetic energy of the system at the instant (some time later)that the blocks are travelling with the same speed? (This is the kineticenergy of the center of mass motion alone.)

d) At this instant, the total compression of the spring is maximum with somemagnitude xmax. Find xmax.

e) Write expressions for x′l(t) and x′

r(t), the position of the left hand andright hand blocks relative to the center of mass of the system.

f) Add these functions to xcm(t) to find xl(t) and xr(t), the position of thetwo blocks as a function of time.


g) Differentiate these solutions to find vl(t) and vr(t), and verify that youranswer obeys the initial condition vl(0) = v0/2, vr(0) = 0. Your overallsolution should describe an “inchworm” crawl of the spring as first onemass momentarily moves at speed v0/2 with the other momentarily atrest, then vice versa.


Problem 401.

problems/oscillation-pr-torsional-oscillator-collision.tex

R

M

θ

x

y

x

y

z

Ω0

τ = −κθz

The torsional oscillator above consists of a disk of mass M and radius Rconnected to a stiff supporting rod. The rod acts like a torsional spring,exerting a restoring torque:

τz = −κθ

if it is twisted through an angle θ counterclockwise around the z axis (see insetabove). κ is the positive torsional spring constant. This torque will makeany object with a moment of inertia that is symmetrically attached to the rodrotationally oscillate around the z axis of the rod as shown.

A second identical disk also of mass M and radius R, rotating around theirmutual axis at an angular speed Ω0, is dropped gently onto the stationaryfirst disk from above and sticks to it (so that they rotate together after thecollision). At the instant of this angular collision, the disks have zero angulardisplacement (i.e. are at the equilibrium angle, θ0 = 0)1.

a) Find the final angular speed Ωf of the two disks moving together imme-diately after the collision (and before the disks have time to rotate).

b) Find the energy that was lost in this (inelastic) rotational collision.

1Note that I’m using a capital omega Ω = dθ/dt to help you keep track of the angularspeed Ω of the disks and angular frequency ω of the oscillator separately below. If youcannot remember the moment of inertia of a disk in terms of M and R, use the symbol Id forthe moment of inertia of a single disk where appropriate in your answers (and lose 2 points).


c) From the torque equation given above, find the differential equation ofmotion for θ(t) for the two disks moving together after the collision.Identify ω2 (the angular frequency of the oscillator after the collision)in this equation, and write down the solution θ(t) in terms of Ωf , κ, Mand R. You do not have to substitute your answer to a) for Ωf .


Problem 402.

problems/oscillation-pr-torsional-oscillator-spring-hard.tex

M

Raxle

k

0 (out)

Frictionless table and axle looking down

x eq

fixed bar

In the figure above, a disk of radius R and mass M is mounted on a nearly fric-tionless axle. A massless spring with spring constant k is attached to a point onits circumference so that it is in equilibrium as shown. The disk is then lightlystruck at time t = 0 so that it is given a small instantaneous counterclockwiseangular velocity of ω0 while it is still at the equilibrium position, and it subse-quently oscillates approximately harmonically through a small maximum angleθ0. Note: Idisk = 1

2MR2 about its center of mass.

a) Find the angular frequency ωf of its oscillation, assuming that the axle isfrictionless and exerts no torque on the disk. (Note well that this is notthe same thing as the initial angular velocity of the disk!)

b) Find the angle θ0 through which it will rotate before (first) coming mo-mentarily to rest in this frictionless case.

c) Suppose that the axle exerts a weak “drag” torque on the disk when thedisk rotates. Do you expect the frequency of oscillation to be larger,smaller, or the same as ωf once drag is taken into account? (Note thatyou do not have to derive an answer, but you should justify it on intuitivegrounds.)

d) Draw a qualitatively correct graph of θ(t), the angle the disk has rotatedthrough (relative to equilibrium) as a function of time when drag/frictionis included as in c).

(Continued workspace on next page)


(Continuation of oscillator problem)


Problem 403.

problems/oscillation-pr-torsional-oscillator-spring.tex

M

Raxle

k

0 (out)

Frictionless table and axle looking down

x eq

fixed bar

In the figure above, a disk of radius R and mass M is mounted on a frictionlessaxle. A massless spring with spring constant k is attached to a point on its cir-cumference so that it is in equilibrium as shown. The disk is then lightly struckat time t = 0 so that it is given a small instantaneous counterclockwise angularvelocity of Ω0 while it is still at the equilibrium position, and it subsequentlyoscillates approximately harmonically through a small maximum angle θ0.

a) Find the angular frequency ω of its oscillation. You may want to obtainthe differential equation of motion first.

b) Find the angle θ0 through which it will rotate before (first) coming mo-mentarily to rest.

c) Write down (or find) θ(t), the angle the disk rotates through (relative toequilibrium) as a function of time.


Problem 404.

problems/oscillation-pr-vertical-bar-and-spring-2.tex

Lθ

kL/3

A uniform vertical bar of mass M and length L is pivoted at the bottom. Aspring with spring constant k is attached a height L/3 over the pivot. This springis strong enough that the bar will oscillate harmonically about the vertical if itis tipped over to a small angle θ and released.

Find:

a) The total torque (magnitude and direction, where θ is positive into thepage as shown) due to both gravity and the spring as a function of θ.

b) What is the angular frequency ω of the bar as it oscillates? Recall thatthe moment of inertia of a uniform bar is 1

3ML2.

c) What is the smallest value that k can have such that the bar is in stableequilibrium in the vertical position? [If the spring constant is larger thanthis smallest value of k, the spring can sustain oscillations of the bar anddoes not fall over if perturbed from equilibrium.]


Problem 405.

problems/oscillation-pr-vertical-bar-and-spring.tex

L/2

L

k

θ

A uniform vertical bar of mass M and length L is pivoted at the bottom. Aspring with spring constant k is attached a height L/2 over the pivot. This springis strong enough that the bar will oscillate harmonically about the vertical if itis tipped over to a small angle θ and released.

Find:

a) The total torque (magnitude and direction, where into the page is pos-itive θ as shown) due to both gravity and the spring as a function ofθ.

b) What is the angular frequency ω of the bar as it oscillates? Recall thatthe moment of inertia of a uniform bar is 1

3ML2.

c) What is the smallest value that k can have such that the bar is in stableequilibrium in the vertical position? [If the spring constant is larger thanthis smallest value of k, the spring can sustain oscillations of the bar anddoes not fall over if perturbed from equilibrium.]


Problem 406.

problems/oscillation-pr-youngs-modulus.tex

AL ∆L

FFY

A cylindrical bar of material with cross-sectional area A, unstressed length L,and a Young’s Modulus Y is subjected to a force F that stretches the bar asshown. The bar behaves like an elastic “spring”, pulling back with a forceF = −k∆L.

a) Show that the “spring constant” of the bar is k = AY/L.

b) Show that the energy stored in the bar when it is stretched by length ∆Lis U = 1

2F∆L. This will be easiest if you assume that the bar is a “spring”with the spring constant k determined in part a).

Chapter 13

Waves

What do you get when you cross an owl with a bungee cord? Ooo, wrongriddle1.

We’ll try again. What do you get when you stretch out a bungee cord and pluckit like an owl?

A wave. A wave is basically what you get when you have a whole bunch ofharmonic oscillators all coupled together so each one can push on its neighbors.Or when one part of a string can pull on its neighbors. Or when one piece ofmatter can push on its neighbor. Or when one studies any of the fields that arethe basic laws of nature. Or when one studies quantum wave mechanics.

But wait! Wave mechanics is chemistry, chemistry is biology, and biology is us.Does that mean that we are, basically, a very complex wave phenomenon?

Damn skippy it does. It also means that if you drill down to the fundamentalsof nearly any science you’re gonna find waves down there, tormenting you withtheir perplexity and complexity, unless you master them now. And understand-ing waves on a humble guitar string, or the wave pulses you can put on a slinkyor garden hose is the very first baby step in that direction.

1Interested parties might want to google up Kung Pow Owl Bungee Cord” or the like andwatch a youtube clip from the movie to learn the highly politically incorrect but hilariousanswer.

463

464 CHAPTER 13. WAVES

13.1 Waves


Problem 407.

problems/waves-mc-fixed-and-free-fundamental.tex

Consider a vibrating string of length L = 2 m which is fixed at one end and freeat the other end. It is found that there are successive standing wave resonancesat 50 Hz and 70 Hz. Then the standing wave with the lowest possible frequency(i.e. the first mode or fundamental mode) has frequency:

a) 10 Hz.

b) 20 Hz.

c) 30 Hz.

d) 40 Hz.

e) 50 Hz.

13.1. WAVES 465

Problem 408.

problems/waves-mc-hanging-string-horizontal-string-speeds.tex

L

m

m

L down

A

B

(6 points) Two strings with identical mass per unit length µ are drawn above.Both are stretched by means of an identical weight of mass m hanging on eachstring as shown. At time t = 0 an identical wave pulse is flipped onto eachstring, where it travels to the end, reflects off of the mass or the pulley as thecase may be, and returns. Which is true:

a) The pulse on string A returns first.

b) The pulse on string B returns first.

c) The pulse on string A returns at the same time as the pulse on string B.


Problem 409.

problems/waves-mc-unknown-fixed-and-free-bcs-fundamental.tex

Consider a vibrating string of length L = 2 m which is fixed at one end and freeat the other end. It is found that there are successive standing wave resonancesat 50 Hz and 70 Hz. Then the standing wave with the lowest possible frequency(i.e. the first mode or fundamental mode) has frequency:

a) 10 Hz.

b) 20 Hz.

c) 30 Hz.

d) 40 Hz.

e) 50 Hz.

13.1. WAVES 467

Problem 410.

problems/waves-mc-unknown-fixed-bcs-fundamental.tex

You are given the following information resulting from measurements of theresonant modes of a string of length L with unknown boundary conditions. Youare told that two successive resonant modes have frequencies of fm = 350 Hzand fm+1 = 400 Hz for some mode index m. Select the true statement from thefollowing list:

a) The fundamental frequency is 50 Hz, the string is definitely fixed at bothends, and m = 7.

b) The fundamental frequency is 50 Hz, the string is definitely free at bothends, and m = 7.

c) The fundamental frequency is 50 Hz, the string is definitely fixed at oneend and free at the other, and m = 4.

d) The fundamental frequency is 50 Hz, the string might be fixed or free atboth ends, and m = 7.

e) The fundamental frequency is 100 Hz, the string might be fixed or freeat both ends, and m = 3.



Problem 411.

problems/waves-ra-wave-energies.tex

a) y(x, t) = A0 sin(k0x − ω0t) Ea = E0

b) y(x, t) = 2A0 sin(k0x − ω0t) Eb =

c) y(x, t) = A0 sin(k0x + ω0t) Ec =

d) y(x, t) = A0 sin(k0x − ω0t + π/6) Ed =

e) y(x, t) = A0 cos(k0x − ω0t) Ee =

f) y(x, t) = A0 sin(k0x − 2ω0t) Ef =

Six strings with the same mass density each have a different traveling har-monic wave on them as given above. Fill in the missing energy per unitlength for each string, in terms of the energy per unit length of the first stringEa = E0.

13.1. WAVES 469

13.1.3 Short Answer

Problem 412.

problems/waves-sa-breaking-guitar-string.tex

A certain guitar string is tuned to vibrate at the (principle harmonic) frequencyf when its tension is adjusted to T . The string will break at a tension 3T .

a) Can you double the frequency of the string by increasing the tension (only)without breaking the string?

b) What is the maximum frequency that you can make the string have, interms of f , without (quite) breaking the string?


Problem 413.

problems/waves-sa-heavy-to-light.tex

v µsmallerµlarger

One end of a heavy rope is tied to a lighter rope as shown in the figure. Anupright wave pulse is incident from then left and travels to the right reachingthe junction between the ropes at time t = 0, so that, for time t > 0, there aretwo pulses - a transmitted pulse in the light rope and a reflected pulse in theheavy rope.

Compare the transmitted and reflected pulses to the incident pulse by fillingin the table below (each answer is “relative to the same property of the incidentpulse”):

Transmitted Reflected

speed (greater, lesser, equal)

orientation (upright, inverted)

energy (greater, lesser, equal)

13.1. WAVES 471

Problem 414.

problems/waves-sa-reflected-wave-energy.tex

A string of some mass density µ is smoothly joined to a string of greater massdensity and the combined string is stretched to a uniform tension Ten (the samein both wires). The speed of a wave pulse on the thinner wire is twice thespeed of a pulse on the thicker wire. A wave pulse reflected from the thin-to-thick junction has half the amplitude of the original pulse. Assuming no lossof energy in the wire:

a) What fraction of the incident energy is reflected at the junction?

b) Is the reflected pulse upside down or right side up?


Problem 415.

problems/waves-sa-reflection-transmission-at-junction.tex

a

b

va

vb

Two combinations of two strings with different mass densities are drawn abovethat are connected in the middle. In both cases the string with the greatest massdensity is drawn darker and thicker than the lighter one, and the strings have thesame tension T in both a and b. A wave pulse is generated on the string pairsthat is travelling from left to right as shown. The wave pulse will arrive at thejunction between the strings at time ta (for a) and tb (for b). Sketch reasonableestimates for the transmitted and reflected wave pulses onto the a and b figuresat time 2ta and 2tb respectively. Your sketch should correctly represent thingslike the relative speed of the reflected and transmitted wave and any changesyou might reasonably expect for the amplitude and appearance of the pulses.

13.1. WAVES 473

Problem 416.

problems/waves-sa-string-fixed-both-ends.tex

T, µy

x

0 L

A string of mass density µ is stretched to a tension T and fixed at x = 0 andx = L. The transverse string displacement is measured in the y direction. Allanswers should be given in terms of these quantities or new quantities (suchas v) you define in terms of these quantities.

Write down kn, ωn, fn, λn for the first three modes supported by the string.Sketch them in on the axes below, labelling nodes and antinodes. You do nothave to derive them, although of course you may want to justify your answersto some extent for partial credit in case your answer is carelessly wrong.

y

x

0 L

y

x

0 L

y

x

0 L


Problem 417.

problems/waves-sa-two-string-densities-frequency.tex

Two identical strings of length L, fixed at both ends, have an identical tensionT , but have different mass densities. One string has a mass density of µ, theother a mass density of 16µ

When plucked, the first string produces a (principle harmonic) tone at frequencyf1. What is the frequency of the tone produced by the second string?

a) f2 = 4f1

b) f2 = 2f1

c) f2 = f1

d) f2 = 12f1

e) f2 = 14f1

13.1. WAVES 475

Problem 418.

problems/waves-sa-wave-facts.tex

Answer the five short questions below:

a) Suppose you are given string A with mass density µA that is stretcheduntil it has tension TA. You are given a second identical string B withtwice the tension, TB = 2TA. What is the speed of a wave vB on string Brelative to vA, the speed on string A?

b) Suppose you are given string A with mass density µA that is stretcheduntil it has tension TA. You are given a second string B with four timesthe mass density of A µB = 4µA but at the same tension. What is thespeed of a wave vB on string B relative to vA, the speed on string A?

c) Suppose you are given string A with mass density µA that is stretcheduntil it has tension TA. You are given a second identical string B withtwice the tension, TB = 2TA. Both strings are carrying a harmonic waveat the same angular frequency ω. What is the wave number kB on stringB in terms of the wave number kA on string A?

d) Suppose you are given string A with mass density µA that is stretcheduntil it has tension TA. You are given a second string B with four timesthe mass density of A µB = 4µA but at the same tension. Both stringsare carrying waves with the same wavelength λ. What is the (regular)frequency fB on string B in terms of the frequency fA on string A?

e) Suppose you are given string A with mass density µA that is stretcheduntil it has tension TA. You are given a second string B with four timesthe mass density of A µB = 4µA and a tension four times the tension of ATB = 4TA. Both strings carry a wave with the same frequency f . Whatis the wavelength λB in terms of the wavelength λA?


Problem 419.

problems/waves-sa-wave-speed-vary-mu.tex

4T 4T

a

b

c

d

µ

2µ

3µ

4µ

In the figure above, the neck of a stringed instrument is schmatized. Four stringsof different thickness and the same length are stretched in such a way that thetension in each is about the same (T ) for a total of 4T between the end bridges– if this were not so, the neck of the guitar or ukelele or violin would tend tobow towards the side with the greater tension. If the velocity of a wave on thefirst (lightest) string is v1, what is the speed of a wave of the other three interms of v1?

13.1. WAVES 477

Problem 420.

problems/waves-sa-wave-speed-vary-T.tex

m 4m 9m

a b c

Three strings of length L (not shown) with the same mass per unit length µ aresuspended vertically and blocks of mass m, 4m and 9m are hung from them.The total mass of each string µL ≪ m (the strings are much lighter than themasses hanging from them). If the speed of a wave pulse on the first string (a)is v0, what is the speed of the same wave pulse on the second (b) and third (c)strings?



Problem 421.

problems/waves-pr-accelerating-wave-pulse.tex

µ

2α tv T(t) = T e0

A wave pulse is started on a string with mass density µ with an applied tensionthat increases like T0e

2αt.

a) Find the initial velocity of the wave pulse at time t = 0.

b) Find the acceleration of the wave pulse as a function of time.

13.1. WAVES 479

Problem 422.

problems/waves-pr-construct-transverse-travelling-wave.tex

A very long string aligned with the x-axis is being shaken at the ends in such away that there is a travelling harmonic wave on it moving in the general xdirection (either positive or negative). y is the vertical direction perpendicularto the string in the direction of the string’s displacement. The amplitude of thewave is 1 cm. The wavelength of the wave is 0.5 meters. The period of the waveis 0.001 seconds.

a) Write down the formula for a transverse wave travelling to the −x direction(left) corresponding to these numerical parameters.

b) What is the speed of the wave on the string?


Problem 423.

problems/waves-pr-fixed-both-ends.tex

A string with mass density µ and under tension T vibrates in the y-direction.The string is fixed at both ends at x = 0 and x = L. Answer all questions interms of these givens.

a) What are the two lowest frequencies f1 and f2 that a standing wavecan have for this string?

b) Write down an equation for y(x, t), the y-displacement of the string as afunction of position x along the string and time t for the standing wavecorresponding to the second lowest frequency f2 (the second mode)that you just computed. Assume that the standing wave has a maximumvertical displacement of y = A.

c) On the graph below, plot the y-displacement for the second mode versushorizontal position x at an instant when the the string achieves its max-imum displacement. Indicate the positions on the x-axis of any nodes orantinodes.

L x

y

−A

A

0

13.1. WAVES 481

Problem 424.

problems/waves-pr-fixed-one-end.tex

A string with mass density µ and under tension T vibrates in the y-direction.The string is fixed at x = 0 and free at x = L. Answer all questions in termsof these givens.

a) What are the two lowest frequencies f1 and f2 that a standing wavecan have for this string?

b) Write down an equation for y(x, t), the y-displacement of the string as afunction of position x along the string and time t for the standing wavecorresponding to the second lowest frequency f2 (the second mode)that you just computed. Assume that the standing wave has a maximumvertical displacement of y = A.

c) On the graph below, plot the y-displacement for the second mode versushorizontal position x at an instant when the the string achieves its max-imum displacement. Indicate the positions on the x-axis of any nodes orantinodes.

L x

y

−A

A

0


Problem 425.

problems/waves-pr-speed-on-hanging-string.tex

L

A string of total length L with a mass density µ is shown hanging from theceiling above.

a) Find the tension in the string as a function of y, the distance up fromits bottom end. Note that the string is not massless, so each small bit ofstring must be in static equilibrium.

b) Find the velocity v(y) of a small wave pulse cast into the string at thebottom that is travelling upward.

c) Find the amount of time it will take this pulse to reach the top of thestring, reflect, and return to the bottom. Neglect the size (width in y) ofthe pulse relative to the length of the string.

13.1. WAVES 483

Problem 426.

problems/waves-pr-standing-wave-mode-energy.tex

0 L

A string of total mass M and total length L is fixed at both ends, stretched sothat the speed of waves on the string is v. It is plucked so that it harmonicallyvibrates in its n = 4 mode:

y(x, t) = A4sin(k4x)cos(ω4t).

Find (derive) the instantaneous total kinetic energy in the string in terms of M ,L, n = 4, v and A4 (although it will simplify matters to use k4 and ω4 onceyou define them).

Remember (FYI):

∫ nπ

0

sin2(u)du =

∫ nπ

0

cos2(u)du =nπ

2


Problem 427.

problems/waves-pr-string-and-hanging-mass.tex

ω

L

m

µ

In the figure above, a string of length L and mass density µ is run over a pulleyand maintained at some tension by a stationary hanging mass m. The string isdriven with tiny oscillations at a tunable frequency ω by a speaker attached toone end as shown. You may neglect the weight of the string compared to theweight of the mass m.

a) For a given mass m, write an expression for the velocity of waves on thestring.

b) Find the frequency of the third harmonic of the string (expressed interms of the givens).

c) What is the angular frequency of the sound wave that the string willproduce when it is driven in resonance with its third harmonic frequency?

d) What is the wavelength of the sound wave produced by the string vibratingat this frequency? You may express your answer algebraically in terms ofva (the speed of sound in air).

13.1. WAVES 485

Problem 428.

problems/waves-pr-travelling-wave-analysis.tex

A travelling wave on a string of mass µ = 0.01 kg/meter is given by the expres-sion:

y(x, t) = 2.0 sin(0.02πx + 2πt) (meters)

Answer the following questions about this wave. All of the arithmetic should

be doable without a calculator, but if you have any doubt feel free to leavearithmetical expressions of the algebra unevaluated.

a) What is the amplitude of this wave?

b) What is its wavelength?

c) What is its period?

d) What is the velocity of this wave (include direction!)?

e) Write an algebraic expression for the kinetic energy per unit length in thestring as a function of time.

Chapter 14

Sound

What’s that again? Did you say something? I couldn’t hear you, because thesound of your voice wasn’t intense enough for me to hear. This isn’t your fault– I’m aging and hence growing deaf(er). And besides, you’re probably not evenin the same room with me.

In any event, understanding sound seems once again like it would be very usefulto physicians, engineers, physicists, musicians, communications specialists, andmaybe even ordinary people who want to learn a bit about waves in what isstill a relatively simple and ubiquitous application before tackling some bit ofscience that has some difficult wave mechanics in it.

In the problems below we will have our first encounter with 3 dimensional waves(the symmetric pretty easy kind), 1/r2 laws, the idea of decibels” and the useof logarithmic scales to describe things that vary over many orders of magni-tude efficiently. We’ll also get our first very tiny taste of interference arisingfrom wave superposition, although the main course is put off until you coverelectromagnetic waves and the interference and diffraction of light waves.

Ping.

487

488 CHAPTER 14. SOUND

14.1 Sound Waves


Problem 429.

problems/sound-mc-car-horn-decibels.tex

You are stuck in freeway traffic and need to get home. So does the driver nextto you – she starts blowing the horn of her car, which you hear as a sound witha sound level of 90 dB. Not to be outdone, the driver behind you, in front ofyou, and to the other side of you all lean on their horn as well, so that nowyou are hearing all four horns (which reach your ears with equal intensities) atonce. The sound level you now hear is:

a) 93 db

b) 96 dB

c) 180 dB

d) 360 dB

e) Unchanged.

14.1. SOUND WAVES 489

Problem 430.

problems/sound-mc-exam-noise-in-dB.tex

200 students are taking an examination in a room, and the sounds of pensscratching on paper, sighs, groans, and muttered imprecations has created amore or less continuous sound level of this noise of 60 dB. Assuming each studentcontributes equally to this noise and nothing else changes or adds to it, whatwill the sound level in the room be when only 50 students are left?

a) 50 dB

b) 15 dB

c) 66 dB

d) 54 dB

e) 57 dB


Problem 431.

problems/sound-mc-measuring-speed-of-sound.tex

water reservoir

tuningfork

test gas

resonating tube

L

hose

microphone

A simple method for measuring the speed of sound in a reservoir filled with gasis to hold a tuning fork at a fixed, known frequency above a tube connected witha flexible hose to a reservoir such that the height of the water in the tube caneasily be varied. The sound one detects with a microphone is then the loudestwhen the tuning fork is in resonance with standing wave modes in the tube.

If you hold a 2000 Hz tuning fork above the tube when it is completely full andthen lower the reservoir slowly to drop the water level in the tube, you hear thefork resonate most loudly when the water is L = 2.5, 7.5, and 12.5 cm beneaththe end of the tube. The speed of sound in the gas is therefore:

a) 50 m/sec

b) 100 m/sec

c) 200 m/sec

d) 500 m/sec

e) 750 m/sec


Problem 432.

problems/sound-mc-shooting-a-gun-dB.tex

A 30-06 rifle makes a bang that peaks at 180 decibels 1 meter away from themuzzle. If you are standing 100 meters away (approximately) what sound leveldo you hear in decibels?

a) 100 dB

b) 120 dB

c) 140 dB

d) 160 dB



Problem 433.

problems/sound-mc-siren-in-dB.tex

A siren radiates sound energy uniformly in all directions. When you stand adistance 100 m away from the siren you hear a sound level of 90 dB. If you moveto a distance of 10 m from the siren, the sound level is:

a) 90 dB, no change.

b) 100 dB.

c) 110 dB.

d) 120 dB.

e) 130 dB.


Problem 434.

problems/sound-mc-sound-level-to-pressure-1.tex

You measure the intensity level of a single frequency sound wave produced bya loudspeaker with a calibrated microphone to be 80 dB. At that intensity, thepeak pressure in the sound wave at the microphone is P0 + Pa, where Pa isthe baseline atmospheric pressure and P0 is the pressure over that associatedwith the wave. The loudspeaker’s amplitude is turned up until the measuredintensity level is 120 dB. What is the peak pressure of the sound wave now?

a) 4P0 + Pa

b) 10P0 + Pa

c) 40P0 + Pa

d) 100P0 + Pa

e) 100(P0 + Pa)


Problem 435.

problems/sound-mc-sound-level-to-pressure-2.tex

You measure the sound level of a single frequency sound wave produced by aloudspeaker with a calibrated microphone to be 80 dB. At that intensity, thepeak pressure in the sound wave at the microphone is P0 + Pa, where Pa is thebaseline atmospheric pressure and P0 is the pressure over that associated withthe wave. The loudspeaker’s amplitude is turned up until the measured soundlevel is 100 dB. What is the peak pressure of the sound wave now?

a) 4P0 + Pa

b) 10P0 + Pa

c) 40P0 + Pa

d) 100(P0 + Pa)

e) 100P0 + Pa



Problem 436.

problems/sound-ra-sound-resonances.tex

d

ba

c

L

L L/2

L

a) Rank the fundamental harmonic resonant frequencies (n = 1) of the fouropen/closed pipes drawn above, where equality is a possible answer. Ananswer might be (but probably isn’t) fa < fb = fc < fd.

b) Draw into each pipe a representation of a the displacement wave asso-ciated with each resonance.

c) Label the nodes (in your representation of the waves) with an N andantinodes with an A.


14.1.3 Short Answer

Problem 437.

problems/sound-sa-beats.tex

Two identical strings of length L have mass µ and are fixed at both ends. Onestring has tension T . The other has tension 1.21T . When plucked, the firststring produces a tone at frequency f1. What is the beat frequency producedif the second string is plucked at the same time, producing a tone f2? Are thebeats likely to be audible if f1 is 500 Hz?


Problem 438.

problems/sound-sa-decibels-sun-human-body.tex

Sunlight reaches the surface of the earth with roughly 1000 Watts/meter2 ofintensity. What is the “intensity level” of a sound wave that carries as muchenergy per square meter, in decibels? In table 15-1 in Tipler and Mosca, whatkind of sound sources produce this sort of intensity? Bear in mind that the Sunis 150 million kilometers away where sound sources capable of reaching the sameintensity are typically only a few meters away. Hmmm, seems as though theSun produces a lot of (electromagnetic) energy compared to terrestrial sourcesof (sound) energy.

While you are at it, the human body produces energy at the rate of roughly100 Watts. Estimate the fraction of this energy that goes into my lecture whenI am speaking in a loud voice in front of the class (loud enough to be heard asloudly as normal conversation ten meters away).


Problem 439.

problems/sound-sa-principle-harmonic-series.tex

Two pipes used in different musical instruments have the same length L, butthe fundamental frequency (frequency of the principal harmonic, m = 1) of oneis twice that of the other. Explain how this could be, illustrating your answerwith a drawing of two pipes and the principle modes such that this is true.


Problem 440.

problems/sound-sa-scaling-thunder-dB.tex

Lightning strikes one kilometer away, and the resulting thunderclap has an in-tensity of 5×10−3 Watts/meter2. What is the intensity level in decibels? If oneis instead 10 kilometers away, approximately how many decibels lower wouldthe intenstity level be?


Problem 441.

problems/sound-sa-scaling-time-thunder-dB.tex

You see the flash of lightning and three seconds later you hear a thunderclapwith a peak sound level of 120 dB. A few minutes later you see a second flashof lightning and twelve seconds later you hear the thunderclap.

a) Approximately what peak sound level do you hear in the second (presum-ably “identically produced”) thunderclap?

Second thunderclap is: dB

b) Roughly – to the nearest kilometer – how far away are the two lightningflashes?

First (three seconds): km

Second (twelve seconds): km


Problem 442.

problems/sound-sa-sound-speed-decibels.tex

(12 points) Some short questions about sound:

a) Show that doubling the intensity of a sound wave corresponds to an in-crease in its intensity level or loundness by about 3 dB.

b) I sometimes work as a timer at my son’s swim meets. We are told tostart our watches when we see a light flash on the starter’s console, notwhen we hear the starting horn. If I am timing a lane on the far side ofthe pool some 17 meters away from the starter and start when the soundof the horn reaches me, how much will the times I measure (on average)change? Will the swimmer have an advantage or a disadvantage relativeto a swimmer timed by someone that starts on the flash of light?

c) Suppose I turn the knob on my surround-sound amplifier and decrease theloudness where I’m listening by 6 dB. By roughly what fraction has theamplitude of oscillation of the speakers changed?


Problem 443.

problems/sound-sa-tube-open-both-ends-1.tex

λ = 1

L L

λ = 2

A tube open at both ends used as a “panpipe” musical instrument. It has lengthL = 34 centimeters.

a) Sketch the first two displacement modes (or harmonics) in the pro-vided tubes.

b) Label the nodes and antinodes, and underneath each tube indicate thewavelength of the mode/harmonic.

c) What is the frequency of the second harmonic of the tube (an actualnumber, please, in Hertz or cycles per second).


Problem 444.

problems/sound-sa-tube-open-both-ends-2.tex

λ = 1

L L

λ = 2

A tube open at both ends used as a “panpipe” musical instrument. It has lengthL = 34 centimeters.

a) Sketch the first two displacement modes (or harmonics) in the pro-vided tubes.

b) Label the nodes and antinodes, and underneath each tube indicate thewavelength of the mode/harmonic.

c) What is the frequency of the principle harmonic of the tube (an actualnumber, please, in Hertz or cycles per second).


Problem 445.

problems/sound-sa-two-wave-speeds-frequency.tex

Two identical pipes, both closed at both ends, are filled with two differentgases. In the first gas, the speed of sound is v1 =

√

B1/ρ1, in the second the

speed of sound is v2 =√

B2/ρ2 = 2v1. Both pipes are driven by speakers inresonance with their fundamental harmonic frequency, f1 and f2 respec-tively.

If f1 is the fundamental frequency in the first pipe, what is the fundamentalfrequency f2 in the second pipe?

a) f2 = 4f1

b) f2 = 2f1

c) f2 = f1

d) f2 = 12f1

e) f2 = 14f1



Problem 446.

problems/sound-pr-bill-and-ted-double-doppler-1.tex

aaaaaaaaaa = fo

Lava

Bill and Ted are falling at a constant speed (terminal velocity) into hell, andare screaming at a frequency f0. They hear their own voices reflecting back tothem from the puddle of molten rock that lies below at a frequency of 2f0. Howfast are they falling relative to the speed of sound?


Problem 447.

problems/sound-pr-bill-and-ted-double-doppler-2.tex

aaaaaaaaaa = fo

Lava

Bill and Ted are falling at a constant speed (terminal velocity) into hell, andare screaming at a frequency f0. They hear their own voices reflecting back tothem from the puddle of molten rock that lies below at a frequency of 1.21f0.How fast are they falling relative to the speed of sound?


Problem 448.

problems/sound-pr-doppler-moving-receiver.tex

s rurv

A microphone mounted on a cart is moved directly toward a harmonic sourceat a speed of 34 m/sec. The harmonic source is emitting sound waves at afrequency of 1360 Hz.

a) Derive an expression for the frequency of the waves picked up by themoving microphone.

b) What is that frequency?


Problem 449.

problems/sound-pr-doppler-moving-source-derive.tex

sv

source receiver

A speaker mounted on a cart is moved directly toward a stationary microphoneat a speed vs = 34 m/sec. It is emitting harmonic sound waves at a sourcefrequency of f0 = 1000 Hz. va = 340 m/sec is the speed of sound in air.

a) Derive an algebraic expression for the frequency f ′ of the waves pickedup by the stationary microphone, beginning with a suitable picture ofthe wave fronts. Limited partial credit will be awarded for just correctlyremembering it if you cannot derive it.

b) What is the frequency f ′ in Hz? You should be able to do the arithmeticwithout a calculator.


Problem 450.

problems/sound-pr-doppler-moving-source.tex

sv

source receiver

A speaker mounted on a cart is moved directly toward a stationary microphoneat a speed vs = 34 m/sec. It is emitting harmonic sound waves at a sourcefrequency of f0 = 1000 Hz. va = 340 m/sec is the speed of sound in air.

a) What is the frequency f ′ of the waves picked up by the microphone inHz? You should be able to do the arithmetic without a calculator.

b) Suppose a second source with the same frequency f0 was located at restan identical distance to the right of the microphone receiver. What wouldbe the frequency of the beats recorded by the microphone?


Problem 451.

problems/sound-pr-lithotripsy-decibels.tex

100 MPa

A = 1 cm 2

kidney stone

Modern lithotripsy machines create a focused acoustical shock wave (SW)pulse with an overpressures that range from P0 = 4×107 to over 108 Pascals1. Aharmonic wave in water with this amplitude would have an intensity I ∼ P 2

0×104

when P0 is expressed in atmospheres of pressure and I is the usual watts persquare meter. Although this expression will not be exact for a non-harmonicshock wave pulse, it should give the right order of magnitude for the averageintensity in the initial peak.

a) Estimate I for an acoustical pulse with a peak amplitude of 108 Pascals.Algebra first! Careful with the units!

b) Express this intensity in decibels. Use the usual reference intensity forsound waves (the threshold of hearing).

c) Estimate the “instantaneous” peak force (rise time on the order of nanosec-onds) exerted by the shock wave overpressure on the front face of a cylin-drical kidney stone with an area of 1 square centimeter.

d) Assuming that this primary pulse lasts for ∆t = 10 nanoseconds (or 10−8

seconds), what is the total impulse imparted to the front face of the kidneystone by this force?

1This dynamic pressure is comparable to the static pressure in the deep ocean trenchesten kilometers beneath the surface, where even “incompressible” water compresses by around4 or 5%.


Problem 452.

problems/sound-pr-standing-waves-organ-pipe.tex

Resonant Sound Waves

Tube closed at one end

An organ pipe is made from a brass tube closed at one end as shown. The pipeis 3.4 meters long. When driven it produces a sound that is a mixture of thefirst, third and sixth harmonic (mode).

a) What are the frequencies of these modes?

b) Sketch the wave amplitudes for the third harmonic mode (only) in on thefigure, indicating the nodes and antinodes. Be sure to indicate whetherthe nodes or antinodes are for pressure/density waves or displacementwaves!

c) The temperature in the church where the organ plays varies by around30 C between summer and winter. By how much (approximately) doesthis vary the frequency of the fundamental harmonic? (Indicate why youranswer is what it is, don’t just put down a guess).


Problem 453.

problems/sound-pr-train-double-doppler-shift-2.tex

A train approaches a tunnel in a sheer cliff at speed vtrain. The train blows awhistle of frequency 1000 Hz. A listener on the train hears a beat frequency of10 Hz between the original whistle and the reflected sound.

a) What is the frequency of the reflected wave as heard by the passengers onthe train?

b) Find the speed of the train relative to the speed of sound in air:

vtrainvair

=


Problem 454.

problems/sound-pr-train-double-doppler-shift.tex

A train approaches a tunnel in a sheer cliff. The train is moving at 34 m/s, andit blows a horn of frequency 900 Hz. The speed of sounds is 340 m/s.

a) What frequency would a listener at the base of the cliff hear?

b) What frequency do the train passengers hear from the echo (the reflectionfrom the cliff face)?

Chapter 15

Gravity

We began this book and the study of mechanics and dynamics with gravity. Itis only fitting that we end it with gravity as well, but this time gravity doneright!

What, the force of gravity is not really mg? Sadly, no, it’s not. This is only itsform near the surface of the earth, at least for the g = 10 m/sec2 we’ve grownto know and love.

Newton’s Law of Gravitation is, at long last, our very first force law of nature,not a composite of other force laws (like normal forces, springs, and so on). It isalso our first inverse square law, and is an excellent preview of Coulomb’s Law(which will start out our studies of electromagnetism shortly).

But the story of gravity is the story of the Universe itself. It is also the story ofthe Enlightenment, the discovery/invention of the scientific method, the system-atic process by which we build up reliable, reproducible knowledge and rejectmythology, lies, errors, fantasies, and just plain (probably) incorrect hypothesesof all sorts. Its importance thus goes far beyond its direct content or directutility.

Physicians probably don’t need to know much about the laws of gravitationother than that it holds us down to the Earth and holds the solar system andgalaxy together. But gravity is so cool that personally, I think everybody

should know a bunch of what is covered in this chapter if only so that they canunderstand a little bit of what they see when they go outside on a cold, clearnight and look up at the stars.

That’s a very human thing to do, and trust me – it is better when you canunderstand a tiny bit of the enormously beautiful structure of what you see.

515

516 CHAPTER 15. GRAVITY

15.1 Gravity


Problem 455.

problems/gravitation-mc-drag-changes-orbit.tex

A satellite in a low-Earth (circular) orbit will slowly lose energy to frictionaldrag forces. What happens to its orbit radius and speed?

a) Its orbit radius increases and its speed increases;

b) Its orbit radius increases and its speed decreases;

c) Its orbit radius decreases and its speed increases;

d) Its orbit radius decreases and its speed decreases;

e) There is no enough information to determine the change to its orbit radiusand speed.

Assume that the orbit remains more or less circular as the drag force acts.Briefly explain or justify your answer.

15.1. GRAVITY 517

Problem 456.

problems/gravitation-mc-escape-velocity-condition.tex

What is the condition for an object near a planetary surface to have escapespeed? Use this condition to derive (in a couple of lines) the escape speed froma planet of mass M and radius R.


Problem 457.

problems/gravitation-mc-kepler-and-scaling.tex

True or False:

a) Kepler’s law of equal areas implies that gravity varies inversely with thesquare of the distance. T F

b) The planet closest to the sun on average (smallest semimajor axis) hasthe shortest period of revolution about the sun. T F

c) The acceleration of an apple near the surface of the earth, compared to theacceleration of the moon as it orbits the earth, is in the ratio of Rm/Re,where Rm is the radius of the moon’s orbit and Re is the radius of theearth. T F

15.1. GRAVITY 519

Problem 458.

problems/gravitation-mc-period-of-missing-planet.tex

The Kepler project is surveying the night sky for stars with planets (and sofar 1800 “exoplanets” have been discovered, with more being found every day).Suppose the Kepler telescope discovers that a gas giant similar to Jupiter (theeasiest kind of planet to detect) is orbiting a particular star at a distance of 4astronomical units (the radius of the Earth’s orbit around the Sun). The periodof the planet’s orbit is determined to be 16 Earth years. What would the periodof a possible Earth-like planet that was orbiting that star at 1 astronomical unitbe?

a) 1/2 Earth year

b) 1 Earth year

c)√

2/2 Earth years

d) 2 Earth years

e) 3 Earth years


Problem 459.

problems/gravitation-mc-surface-gravity-scaling.tex

R 2R R

ρρ

2ρ

m m m

(a) (b) (c)

In the figure above, a small mass m is sitting on the surface of three planets.The density and radius of the planets are as shown:

a) ρ, R

b) ρ, 2R

c) 2ρ, R

If the force on m due to gravity for the first planet is Fa, find and express Fb

and Fc in terms of Fa.

Fb =

Fc =

15.1. GRAVITY 521


Problem 460.

problems/gravitation-ra-four-planets.tex

M,R

M,2R

M,R/2c (hollow)

M,3R/2d

a

b

m,3R m,3R m,3R m,3R

(6 points) Four planets of mass M are drawn to scale above, each exerting agravitational force of magnitude Fi (for i = a, b, c, d) on the small mass m at theposition 3R from the center of each planet as shown. Rank the Fi from least togreatest including possible equalities. Indicate why you are answering the waythat you answer in words or an equation or two.


15.1.3 Short Answer

Problem 461.

problems/gravitation-sa-circular-orbit-vs-escape.tex

It is very costly (in energy) to lift a payload from the surface of the earth intoa circular orbit, but once you are there, it only costs you that same amountof energy again to get from that circular orbit to anywhere you like – if youare willing to wait a long time to get there. Science Fiction author Robert A.Heinlein succinctly stated this as: “By the time you are in orbit, you’re halfwayto anywhere.”

Prove this by comparing the total energy of a mass:

a) On the ground. Neglect its kinetic energy due to the rotation of the Earth.

b) In a (very low) circular orbit with at radius R ≈ RE – assume that it isstill more or less the same distance from the center of the Earth as it waswhen it was on the ground.

c) The orbit with minimal escape energy (that will arrive, at rest, “at infin-ity” after an infinite amount of time).

15.1. GRAVITY 523

Problem 462.

problems/gravitation-sa-force-and-torque.tex

P

O x

y

R

R

M

m

(z out of page)

In the figure above, a mass M is located at the origin, and a mass m is locatedat (0, R) as drawn. The z-axis in the figure comes out of the page. All vectoranswers below may be indicated in any of the permissible ways.

a) Find the gravitional force acting on mass little m.

b) Find the torque around the origin O.

c) Find the torque on mass m relative to the pivot P . Draw and label anarrow symbol onto the figure above to explicitly indicate its direction


Problem 463.

problems/gravitation-sa-identify-four-orbits.tex

E

E

0

1

E2

E3

r

effU

The effective radial potential of a planetary object of mass m in an orbit arounda star of mass M is:

Ueff(r) =L2

2mr2− GMm

r

(a form you already explored in a previous homework problem). The totalenergy of four orbits are drawn as dashed lines on the figure above for some givenvalue of L. Name the kind of orbit (circular, elliptical, parabolic, hyperbolic)each energy represents and mark its turning point(s).

15.1. GRAVITY 525

Problem 464.

problems/gravitation-sa-kepler-3-circular-orbits.tex

In your homework, you studied several different cases of a mass m in a circu-lar orbit around (or inside) another mass M , with different radial force laws.Suppose you are given a radial force law of the form:

~F = − A

rnr

Prove that (for circular orbits in particular):

rn+1 = CT 2

where T is the period of the orbit and r is the radius of the circle, and findthe constant C. (A = GMm, n = 2 then leads to Kepler’s third law, andA = GMm/R3, n = −1 leads to the relation you derived for the mass in thetunnel through the death star).


Problem 465.

problems/gravitation-sa-period-of-saturn.tex

The earth’s orbit is “one astronomical unit” (AU) in radius (this turns out tobe about 150 million kilometers). The period of its orbit is one year. Themean radius of Saturn’s orbit is (roughly) 10 AU. What is its “year” (period ofrevolution around the sun) in years?

15.1. GRAVITY 527

Problem 466.

problems/gravitation-sa-speed-of-jupiter.tex

The Earth’s approximately circular orbit about the Sun is “one astronomicalunit” (AU) in radius (this turns out to be about 150 million kilometers). Themean radius of Jupiter’s approximately circular orbit is (roughly) 5 AU. What isthe average speed of Jupiter vjupiter in terms of the average speed of the Earthvearth as it moves around the Sun?

vjupiter =


Problem 467.

problems/gravitation-sa-two-orbits-scaling.tex

ba

R

2R

Two satellites are in circular orbits around the earth, one at radius R and theother at 2R.

a) Circle the satellite that is moving faster.

b) How much faster is it moving? (Express the faster satellite’s speed interms of the speed of the slower satellite.)

15.1. GRAVITY 529


Problem 468.

problems/gravitation-pr-cavendish-torsional-oscillator.tex

m

m

equilibriumorientation

θ

τ

z

L

torsional thread

restoring torque

In the Cavendish experiment, the gravitational force is measured between twobig masses M (not shown) acting on two small masses m on a rod of lengthL (assumed to be of negligible mass in this problem, although it isn’t really)attached to a thin thread such that it makes a torsional pendulum (as drawnabove). The twisting thread exerts a restoring torque of magnitude τ = −κθon the rod connecting the small masses, where theta is measured from theequilibrium angle of the rod as shown.

In the experiment the two large masses are placed symmetrically so that theyexert a torque on the small mass arrangement aligned with the torsional thread.The two small masses twist the thread toward the big masses until the grav-itational torque is balanced by the torque of the thread. If κ is known, ameasurement of the angle of deviation θ0 suffices to determine the gravitionaltorque, hence the gravitational force, hence the gravitational constant G.

There’s only one catch – one needs κ, and most spools of thread don’t comelabeled with their torsional response properties.

Show and tell how you can do a simple experiment to measure κ with nothing butan accurate stop watch, a measurement of the mass(es) m, and a measurementof the length L of the connecting rod. (Describe the experiment and derive therelation between the quantity you choose to measure and the desired result, κ).


Problem 469.

problems/gravitation-pr-dangerous-tides.tex

rf

rc

m(feet) = 2 kgcenter of mass

Neutron Star M = 10 kg30

Tides can be dangerous. You are a scientist in orbit around a neutron star witha mass M = 1030 kg and a radius of 8 km. Your center of mass moves in aperfect circle 10 km around the center of the star. You have just enough angularmomentum that your feet always point “down” toward the center of the starand your head points away. Your feet are therefore also in a circular trajectoryaround the center of the star, but they cannot also be in orbit (free fall).

Assuming that your feet have a mass of approximately 2 kg and are locatedapproximately 1 meter closer to the star than your center of mass, how muchforce do your legs have to provide to keep your feet from falling off? Do theyfall off?

Hints: Proceed by finding the centripetal acceleration/force of your center ofmass in terms of the gravitational field/force of the star at that location. Repeatthis for your feet separately, assuming that they have the same angular frequencyof circular motion as your center of mass but are in a (much!) stronger gravi-tational field. The difference in the force required to keep the feet in a circularorbit (the total centripetal force) and the actual gravitational force must beprovided by your legs. Also, the binomial expansion might well be useful here...

15.1. GRAVITY 531

Problem 470.

problems/gravitation-pr-dinosaur-killer-asteroid.tex

Earth

Asteroid

Estimate the total energy released when a spherical “Dinosaur Killer” asteroidwith a density ρ = 10 kg/m3 and radius R = 1000 meters falls onto the surfaceof the earth from “outer space” (far away). Obviously your answer should bejustified by a good physical argument.

Note that this is a lot of energy – more than enough to wipe out all life withinperhaps 1000 km of the point of impact (or more) and to change the climate ofthe planet.


Problem 471.

problems/gravitation-pr-escape-velocity-linear-accelerator.tex

M

100 km

F

ev

One way to reduce the cost of lifting mass into orbit is to use a linear acceleratorto drive a payload up to escape velocity (or thereabouts) and then let it go. Thisway one doesn’t have to lift the fuel used to lift the fuel used to lift the . . . (almostall the fuel used in a rocket is used to lift fuel, not payload).

Assume that fusion energy has been developed and electricity is cheap, and thathigh temperature superconductors have made such a mass driver feasible. Yourjob is to do a first estimate of the design parameters.

A proposed plan for the mass driver is shown above. The track is 100 kilometerslong and slopes gently upwards. The payload capsule has a mass of 2 × 103 kg(two metric tons). The head of the track is high in the Andes, R = 6375kilometers from the center of the earth.

a) Neglecting air resistance, find the escape velocity for the capsule. Al-though bound orbits will not require quite as much energy, air resistancewill dissipate some energy. Either way, this is a reasonable estimate of thevelocity the driver must be able to produce.

b) Assuming that the capsule is started from rest and that a constant tan-gential force accelerates it, find the tangential force necessary to achieveescape velocity at the end of the track. Note: Ignore the normal force thatthe track must exert to divert it so that it departs at an upward angle.)From this find the acceleration of the capsule, in multiples of g. Is thisacceleration likely to be tolerable to humans?

15.1. GRAVITY 533

Problem 472.

problems/gravitation-pr-escape-velocity-neutron-star.tex

Neutron Star

comet

Cool stuff about gravity. A neutron star has a mass M = 1030 kg and a radiusR of 8 km. Answer the following problems algebraically using the variablesM , m, G, R first, then (if you have a calculator handy or can do the arithmeticby hand) do the arithmetic and put down numbers. You can get full credit fromthe algebra alone, but the number answers are pretty interesting.

a) What is the escape velocity from the surface of the neutron star? (If youdo the arithmetic, express the result as a fraction of c, the speed of light:c = 3 × 108 m/sec).

b) A comet with a mass m = 1014 kg falls from infinity into the neutron star.What is the energy liberated as it (inelastically) hits?

c) Compare this energy to the total mass energy of the comet, mc2.


Problem 473.

problems/gravitation-pr-escape-velocity-of-baseball.tex

ρ = _____V

M

R

Planetary rock has an average density around ρ = 104 kg/m3. Assuming thatyou can throw a fastball around v = 40 m/sec (nearly 90 mph) what is theapproximate radius R of the largest spherical planet where you can stand onthe surface and throw a baseball away (to infinity)? Note that you can answerthis problem with an algebraic expression and get full credit – the arithmetic isprimarily there so you can learn the answer for yourself!

15.1. GRAVITY 535

Problem 474.

problems/gravitation-pr-geosync-orbit.tex

The Duke Communications company wants to put a satellite into a circulargeosynchronous orbit over the equator (this is a satellite whose period is exactlyone day, so that it stays over the same point of the rotating Earth).

Ignoring perturbations like the Moon and the Sun, find the radius Rg of suchan orbit as a multiple of the radius of the Earth Re. Although as always youshould solve for the result algebraically first you may wish to know some of thefollowing data: The radius of the Moon’s orbit is Rm = 384, 000 kilometers, orRm = 60Re. The period of the Moon is Tm = 27.3 days compared to Tg = 1day. Re = 6400 kilometers. Me = 6 × 1024 kilograms. One day contains 86400seconds.


Problem 475.

problems/gravitation-pr-half-tunnel-escape.tex

rm

Rρ

In the figure above a spherical planet of uniform density ρ and total radius Ris shown. A small tunnel is drilled from the surface to the center.

a) Find the magnitude of the gravitational field g(r) in the tunnel as a func-tion of r.

b) How much work is required to lift a mass m at a constant speed from thecenter of this planet to the surface?

c) Suppose the mass m has reached the surface of the planet and is at rest.What upward-directed speed must you give the mass m at the surface sothat the mass escapes from the planet altogether?

15.1. GRAVITY 537

Problem 476.

problems/gravitation-pr-planet-with-spherical-hole.tex

R

R/2

Above is pictured a spherical mass with radius R and mass density (mass perunit volume) ρ. It has a spherical hole cut out of it of radius R/2 as shown.Find the gravitational field in the hole in terms of G, R, and ρ, proving that itis uniform and points to the left.


Problem 477.

problems/gravitation-pr-spherical-cow.tex

There is an old physics joke involving cows, and you will need to use its punchlineto solve this problem.

A cow is standing in the middle of an open, flat field. A plumb bob with amass of 1 kg is suspended via an unstretchable string 10 meters long so thatit is hanging down roughly 2 meters away from the center of mass of the cow.Making any reasonable assumptions you like or need to, estimate the angle ofdeflection of the plumb bob from vertical due to the gravitational field of thecow.

15.1. GRAVITY 539

Problem 478.

problems/gravitation-pr-thick-shell-force.tex

b

a

ρ

m

r

A “thick” shell of mass with uniform mass density ρ, inner radius a, and outerradius b is shown. A small (frictionless) hole has been drilled at the top alongthe z axis, and a mass m is at a distance r from the center of the shell along thez axis so that it can be moved vertically up or down from outside of the shellto the inside by means of the tunnel.

Find an expression for the magnitude of the radial force Fr acting on m whenthe mass is:

a) Outside of the shell of mass entirely, at some r > b.

b) In the tunnel, where a < r < b.

c) Inside the shell, at some point r < a.


Problem 479.

problems/gravitation-pr-tides-moe-and-joe.tex

2d

R

M

moe

joe

R+d

R−dvc

black hole

Moe and Joe, who have identical masses m, are in a circular orbit around ablack hole about the size of a marble, which contains roughly the same massM as the earth, in the orientation shown above. The radius of the orbit of theircenter of mass is R (which we’ll assume is much larger than the BH). Moe andJoe and tied with a very strong rope 2d meters long (with d ≪ R) that keepsthem moving around the Black Hole at the same angular speed as their centerof mass. Alas, this means that neither Joe nor Moe are actually in orbit (freefall) so the rope has to exert a force to keep them moving with their center ofmass. Find:

a) The speed vc of their center of mass in the circular orbit, as well as itsangular speed ωc, as a function of G, M , and R. This is just an ordinarycircular orbit problem, don’t make it overcomplicated.

b) If Joe (closer to the BH) is moving in a circular trajectory with radiusR−d and the same angular velocity that you obtained in a) as the orbitalangular velocity correct for radius R, what is the net force that must beexerted on Joe by the BH and the rope together?

c) What is the force exerted on Joe by the BH alone at this radius?

d) Therefore, what must the tension T be in the rope (still as a function ofG, M , m, R and d)?

This “force” (opposed by the tension T ) is the tide.

15.1. GRAVITY 541

Problem 480.

problems/gravitation-pr-tunnel-through-death-star-orbit.tex

r

RM

ω


• Newton’s Law of Gravitation

• Circular Orbits

• Centripetal Acceleration

• Kepler’s Laws


A straight, smooth (frictionless) transit tunnel is dug through a spherical aster-oid of radius R and mass M that has been converted into Darth Vader’s death

star. The tunnel is in the equatorial plane and passes through the center of thedeath star. The death star moves about in a hard vacuum, of course, and thetunnel is open so there are no drag forces acting on masses moving through it.

a) Find the force acting on a car of mass m a distance r < R from the centerof the death star.

b) You are commanded to find the precise rotational frequency of the deathstar ω such that objects in the tunnel will orbit at that frequency andhence will appear to remain at rest relative to the tunnel at any pointalong it. That way Darth can Use the Dark Side to move himself alongit almost without straining his midichlorians. In the meantime, he isreaching his crooked fingers towards you and you feel a choking sensation,so better start to work.


c) Which of Kepler’s laws does your orbit satisfy, and why?

15.1. GRAVITY 543

Problem 481.

problems/gravitation-pr-tunnel-through-planet-oscillator.tex

rR

ρ0

North Pole

m

A straight, smooth (frictionless) transit tunnel is dug through a planet of radiusR whose mass density ρ0 is constant. The tunnel passes through the center ofthe planet and is lined up with its axis of rotation (so that the planet’s rotationis irrelevant to this problem). All the air is evacuated from the tunnel toeliminate drag forces.

a) Find the force acting on a car of mass m a distance r < R from the center ofthe planet.

b) Write Newton’s second law for the car, and extract the differential equationof motion. From this find r(t) for the car, assuming that it starts at r0 = R onthe North Pole at time t = 0.

c) How long does it take the car to get to the center of the planet starting fromrest at the North Pole? How long does it take if one starts half way down tothe center? Comment.

All answers should be given in terms of G, ρ0, R and m (or in terms of quantitiesyou’ve already defined in terms of these quantities, such as ω).


Problem 482.

problems/gravitation-pr-two-densities-difficult.tex

rm

R/2

R2ρ

ρ

In the figure above a spherical planet of total radius R is shown that has aspherical iron core with radius R/2 and density 2ρ surrounded by a (liquid)rock mantle with density ρ.

a) Find the gravitational field ~g(r) as a function of the distance from thecenter.

b) Suppose a small, well-insulated tunnel were drilled all the way to thecenter. How much work is required to lift a mass m from the center tothe surface?

15.1. GRAVITY 545

Problem 483.

problems/gravitation-pr-two-spherical-shells.tex

m

2R

M1 M 2

R

A hollow spherical mass shell of mass M1 and radius R is inside another hollowspherical mass shell of mass M2 and radius 2R. The shells are concentric andof negligible thickness.

a) A small mass m is placed on the outer surface of the bigger shell M2.Calculate its acceleration due to gravity g2 in terms of the shell massesM1 and M2, G and R.

b) The mass m is placed on the outer surface of the smaller shell M1. Itsacceleration due to gravity g1 is measured and found to be the same asthe value of g2 from part (a). Use this equality of g1 and g2 to express M2

in terms of M1, G and R.

c) With the relationship you have just derived between M1 and M2, computethe gravitational potential energy P1 of a mass m on the outer surface ofthe bigger shell. Express P1 in terms of G, m, M1 and R, using the con-vention that the gravitation potential is defined as zero at infinite radius.

d) Compute the change in gravitational potential energy ∆P as the mass mmoves from its position on the outer surface of M2 to a position on theouter surface of of M1 (being lowered through the small hole in the outershell). Is the potential energy larger (more positive) at R or 2R?

e) If an object is dropped from rest through the hole in the bigger shell, whatis its speed when it hits the smaller shell? You may give this answer interms of ∆P so that you can get it right even if you get (d) wrong.


Problem 484.

problems/gravitation-pr-vector-field-two-masses.tex

m x

y

M = 80m

D = 5d

d

The large mass above is the Earth, the smaller mass the Moon. Find thevector gravitational field acting on the spaceship on its way from Earth toMars (swinging past the Moon at the instant drawn) in the picture above.

Chapter 16

Leftovers

16.1 Leftover Short Problems

Problem 485.

problems/short-ball-in-circle-on-table.tex

r

m

m1

2

v

A hockey puck of mass m1 is tied to a string that passes through a hole in africtionless table, where it is also attached to a mass m2 that hangs underneath.The mass is given a push so that it moves in a circle of radius r at constantspeed v when mass m2 hangs free beneath the table. Find r as a function ofm1, m2, v, and g.

547

548 CHAPTER 16. LEFTOVERS

Problem 486.

problems/short-coriolis-force.tex

(8 points) You are in a sealed room with no windows or doors. Your apparentweight is approximately normal. A plumb bob hangs straight down from theceiling. However, when a ball bearing made of the same material as the plumbbob is dropped from the ceiling to the floor from a point near the plumb line,it appears to curve a bit away from the line and strikes the floor at a pointdisplaced from “directly beneath” the point from which it fell. Which of thefollowing explanations for this are reasonable and consistent with physics as wecurrently understand it (circle all that might be correct):

a) The room is on a space station, and ”gravity” is produced by the station’srotation in the opposite direction of the deflection.

b) The room is on a train moving in a straight line perpendicular to gravity,which is accelerating uniformly in the direction opposite the deflection.

c) The room is on a rapidly rotating planet – down is down due to gravity, butthe deflection is due to rotational motion into the direction of deflection.

d) Gandalf is in the room next door and cast a spell that deflected the ballbearing.

e) Congratulations! You’ve just discovered irrefutable proof of a fifth forceof nature!

16.1. LEFTOVER SHORT PROBLEMS 549

Problem 487.

problems/short-doppler-detection-of-extrasolar-planets.tex

The Doppler shift of light waves is an important tool (probably the most im-portant tool) used in the search for extrasolar planets, which are currentlybeing discovered at the rate of better than one a day by special telescopes andprojects such as the Kepler project. Explain how this works using the idea ofthe Doppler shift itself plus pictures and physics concepts that are familiar toyou. You are encouraged to use the Internet and talk about this with yourpeers to help get a handle on the idea if it isn’t immediately obvious to you. Atthe moment, with well over 500 extrasolar planets catalogued and more beingdiscovered every week, it is becoming pretty clear that planets are common inour galaxy about stars that contain elements heavier than hydrogen and helium(which are usually “second generation stars”) rather than being in any sensethe exception.

This, in turn, makes it more plausible that life is not unique to our planetor even our solar system, as there are very probably many stars with planetsthe right size, orbital distance from the star, and general chemical environmentrequired to sustain life in the over 100 billion stars of our Milky Way galaxy– and then there are the 500 billion or so other galaxies visible to the HubbleSpace Telescope (with no boundary in sight and no reason to think that therearen’t many trillions of galaxies, each with perhaps a hundred billion stars onaverage, containing quintillions of planets). That’s a whole lot of rolls of thecosmic dice that can lead to an Earth-like planet and life.


Problem 488.

problems/short-draw-Pav-resonance.tex

ω0

Pav

ω

On the axes provided above, draw two qualitatively correct resonance curves forPav(ω), the average power delivered to a damped, driven harmonic oscillator,one for Q = 4 and one for Q = 10. The (labelled!) curves must correctly andproportionately exhibit ∆ω, the full width at half max. Be sure to indicate thealgebraic relation between Q and ∆ω you use to draw the curves.


Problem 489.

problems/short-fermats-principle.tex

1

2

3

A B DC

a

b

Light does not always travel in a straight line between two points. In fact, lighttends to take the path that takes the least amount of time to get between twopoints – this is known as Fermat’s Principle and using it one can understandSnell’s Law next semester, which is the physical principle underlying how lenses(and hence the human eye) work!

In the figure above, the speed of light in the media layers is v1 > v3 > v2. Whichof the labelled pathways is a plausible way for light to travel between points aand b?


Problem 490.

problems/short-graph-latent-heat-vaporization-fusion.tex

−20 0 20 40 60 80 100 120 T

Q

Draw a qualitatively correct graph of the heat that must be added to a kg of iceat -20 C to turn it into water vapor at +120 C as a function of temperature.


Problem 491.

problems/short-harmonic-energy.tex

A mass mA is connected to a spring of spring constant k. The mass is pulledout to a displacement of XA from its equilibrium position and then released. Asecond identical mass mB is connected to an identical spring with the samespring constant k and is pulled out to a displacement of XB = 2XA and released.

Let KA,B be the maximum kinetic energy of each mass respectively as itoscillates, and TA,B be the period of the oscillation of each mass. Circle thetrue statement below:

a) KB = 4KA and TB =√

TA.

b) KB = 4KA and TB = TA.

c) KB = 4KA and TB = 2TA.

d) KB = 2KA and TB = 2TA.

e) KB = 2KA and TB = TA.


Problem 492.

problems/short-hyperbolic-orbit.tex

Many comets are not bound to the sun – they fall in from infinity, make a singlepass by the sun, and then fly out to infinity once again with a positive totalenergy. What is the name of this unbound orbit?


Problem 493.

problems/short-importance-of-oscillation.tex

Harmonic oscillation is conceptually very important because (as has been re-marked in class) many things that are stable will oscillate more or less harmon-ically if perturbed a small distance away from their stable equilibrium point.Draw a graph of a “generic” interaction potential energy with a stable equi-librium point and explain in words and equations where, and why, this shouldbe.


Problem 494.

problems/short-is-anharmonic-force-conservative.tex

Use the definition of a conservative force to determine whether or not the fol-lowing force law (for a one-dimensional anharmonic oscillator) is conservative:

F (x) = −kx − bx3


Problem 495.

problems/short-K2-for-radial-force.tex

Physicists are working to understand “dark matter”, a phenomenological hy-pothesis invented to explain the fact that things such as the orbital periodsaround the centers of galaxies cannot be explained on the basis of estimatesof Newton’s Law of Gravitation using the total visible matter in the galaxy(which works well for the mass we can see in planetary or stellar context). Byadding mass we cannot see until the orbital rates are explained, Newton’s Lawof Gravitation is preserved (and so are its general relativistic equivalents).

However, there are alternative hypotheses, one of which is that Newton’s Lawof Gravitation is wrong, deviating from a 1/r2 force law at very large distances(but remaining a central force). The orbits produced by such a 1/rn force law(with n 6= 2) would not be elliptical any more, and r3 6= CT 2 – but would theystill sweep out equal areas in equal times? Explain.


Problem 496.

problems/short-lifting-card.tex

A card bent at the ends is placed on a table as shown above. Can you blow thecard over by blowing underneath it? (Hint – you can try this at home if youhave a piece of thin cardboard in your dorm.)


Problem 497.

problems/short-math-binomial-expansion.tex

Write down the binomial expansion for the following expressions, given theconditions indicated. FYI, the binomial expansion is:

(1 + x)n = 1 + nx +n(n − 1)

2!x2 +

n(n − 1)(n − 2)

3!x3 + . . .

where x can be positive or negative and where n is any real number and onlyconverges if |x| < 1. Write at least the first three non-zero terms in the expan-sion:

a) For x > a:1

(x + a)2

b) For x > a:1

(x + a)3/2

c) For x > a:(x + a)1/2

d) For x > a:1

(x + a)1/2− 1

(x − a)1/2

e) For r > a:1

(r2 + a2 − 2ar cos(θ))1/2


Problem 498.

problems/short-math-elliptical-trajectory.tex

The position of a particle as a function of time is given by:

~x(t) = x0 cos(ωt)x + y0 sin(ωt)y

where x0 > y0.

a) What is ~v(t) for this particle?

b) What is ~a(t) for this particle?

c) Draw a generic plot of the trajectory function for the particle. What kindof shape is this? In what direction/sense is the particle moving (indicatewith arrow on trajectory)?

d) Draw separate plots of x(t) and y(t) on the same axes.


Problem 499.

problems/short-math-simple-series.tex

Evaluate the first three nonzero terms for the Taylor Series for the followingexpressions. Recall that the radius of convergence for the binomial expansion(another name for the first taylor series in the list below) is |x| < 1 – this givesyou two ways to consider the expansions of the form (x + a)n.

a) Expand about x = 0:

(1 + x)−2 ≈

b) Expand about x = 0:

ex ≈

c) For x > a (expand about x or use the binomial expansion after factoring):

(x + a)−2 ≈

d) Estimate 0.91/4 to within 1% without a calculator, if you can. Explainyour reasoning.


Problem 500.

problems/short-math-sum-two-vectors-1.tex

Suppose vector ~A = −4x + 6y and vector ~B = 9x + 6y. Then the vector~C = ~A + ~B:

a) is in the first quadrant (x+,y+) and has magnitude 17.

b) is in the fourth quadrant (x+,y-) and has magnitude 12.

c) is in the first quadrant (x+,y+) and has magnitude 13.

d) is in the second quadrant (x-,y+) and has magnitude 17.

e) is in the third quadrant (x-,y-) and has magnitude 13.


Problem 501.

problems/short-math-taylor-series.tex

Evaluate the first three nonzero terms for the Taylor series for the followingexpressions. Expand about the indicated point:

a) Expand about x = 0:(1 + x)n ≈

b) Expand about x = 0:sin(x) ≈

c) Expand about x = 0:cos(x) ≈

d) Expand about x = 0:ex ≈

e) Expand about x = 0 (note: i2 = −1):

eix ≈

Verify that the expansions of both sides of the following expression match:

eiθ = cos(θ) + i sin(θ)


Problem 502.

problems/short-momentum-conservation-two-masses-spring-mc.tex

m1 m2

Two masses, m2 = 2m1 are separated by a compressed spring. At time t = 0they are released from rest and the (massless) spring expands. There is nogravity or friction. As they move apart, which statement about the magnitudeof each mass’s kinetic energy Ki and momentum pi is true?

a) K1 = K2, p1 = 2p2

b) K1 = 2K2, p1 = p2

c) K1 = K2/2, p1 = p2/2

d) K1 = K2, p1 = 2p2

e) K1 = K2/4, p1 = p2


Problem 503.

problems/short-momentum-conservation-two-masses-spring.tex

m1 m2

Two masses, m1 < m2 are separated by a compressed spring. At time t = 0 theyare released from rest and the (massless) spring expands. There is no gravityor friction. As they move apart, at all times:

a) Do the two masses have the same speed?

b) Do they have the same kinetic energy?

c) Do they have the same magnitude of momentum?


Problem 504.

problems/short-N3-and-acceleration.tex

Aristotle is walking along in the Elysian Fields having a philosophical argumentwith Newton. Aristotle says to Isaac: “Seriously, Monkey-boy. This third lawof yours, it’s simply ridiculous. For every force acting in a problem there mustbe an equal and opposite force acting in the problem. Why, any fool can seethat if this were so, nothing could move, or whatever you call it, “accelerate”according to your second law!”

“Not so, Moose-breath,” replies Newton (note the fondness with which the twofriends address one another). “It is true for every step that you take and ifyou will just slow to a stop here beneath this tree for a moment I will gladlyexplain...”

How does Newton convince Aristotle that even though all forces occur in pairs,it doesn’t mean that objects cannot accelerate? A picture or two and a simpleargument, please.


Problem 505.

problems/short-orbit-questions.tex

Which of the following statements about gravity and orbits are true? Circle thecorrect ones, and correct the incorrect ones by changing a few words.

a) Planets always move about the Sun in circular orbits.

b) Comets orbiting the sun travel faster when they are farther away thanthey do when they are nearer to the sun.

c) The period of Jupiter’s orbit cubed is proportional to the mean radius ofits orbit squared.

d) If Rm is the distance from the center of the Earth to the center of themoon and Re is the distance from the center of the Earth to a fallingapple, then it is true that the ratio of their accelerations is:

amoon

aapple=

R2e

R2m

e) The Catholic Church prosecuted Galileo for heresy because he claimedthat the Sun orbited the Earth, which contradicted the Biblical model ofcreation.


Problem 506.

problems/short-parallel-series-springs.tex

m

m

mk

k 1 k 2

k 1

k 2

eff

In the figure above a mass M is attached to two springs k1 and k2 in “series”– one connected to the other end-to-end – and in “parallel” – the two springsside by side. In the third picture the same mass is shown attached to a singlespring with constant keff . What should keff be so that the force on the mass isthe same for any given displacement ∆x from equilibrium?


Problem 507.

problems/short-particle-moving-in-circle.tex

A particle of mass m is attached to a string and moving in a vertical circleof radius r at a point in its orbit where its instantaneous speed v. Circle allunambiguously true statements below.

a) Its total acceleration is precisely v2

r towards the center of the circle.

b) The tension in the string must have magnitude m v2

r .

c) The particle must be travelling at a constant speed.

d) There is not enough information to determine any component of its totalacceleration.

e) There is not enough information to determine the total force acting onthe particle.


Problem 508.

problems/short-pendulum-on-the-moon.tex

A pendulum with a string of length L supporting a mass m on the earth hasa certain period T . A physicist on the moon, where the acceleration near thesurface is around g/6, wants to make a pendulum with the same period. Whatmass and length of string could be used to accomplish this?


Problem 509.

problems/short-period-of-asteroid.tex

The earth’s orbit is “one astronomical unit” (AU) in radius (this turns out to beabout 150 million kilometers). The period of its orbit is one year. An asteroid(Hektor) has a nearly circular orbit with a radius of (roughly) 5 AU. What isthe period of Hektor’s revolution around the sun, in years?


Problem 510.

problems/short-planets-around-distant-star.tex

New techniques and improved instrumentation have led to an explosion in thedetection of exoplanets (planets orbiting stars other than the sun) since the firstexoplanet was detected in 1988, with new planets being discovered nearly everymonth. Exoplanets can be detected by (for example) observing tiny variationsin the light emitted by a star when a planet transits across its surface andblocks some fraction of its light. One important piece of data obtained fromthese observations is the period T of the planet’s orbit. Let’s see what we candeduce from this data.

a) Suppose two planets are detected orbiting a certain star. The period ofthe first is determined to be T1 = 1 year. The period of the second isdetermined to be T2 = 9 years. What is the ratio of (the semimajor axisof) their orbits R2

R1

?

b) By means of black magic involving chickens, black handled knives, andcareful observations of the spectrum and intensity of the parent star, it isdetermined that the star has a mass of approximately twice the mass ofour own Sun. In this case, what is the radius of the orbit of the planetwith the period of one year? (Hint: Use the relationship you derived forthe constant of proportionality in Kepler’s Third Law for a circular orbitand give the answer as a fraction of Earth’s orbital radius if you like.)


Problem 511.

problems/short-point-of-N1.tex

Newton’s First Law is a bit “odd”, because one can trivially derive it fromNewton’s Second Law. Which of the following are reasons Newton might havehad for writing it down as a Law on its own (there can be more than one correctanswer):

a) Newton wanted to show people that his new physics was very much likeAristotle’s physics so that they would be more likely to accept it.

b) It permits one to identify inertial reference frames as being any framewhere an object remains in uniform motion if no (net) forces of nature acton it.

c) Because objects at rest remain at rest unless acted on by a nonzero sumof forces of nature in all reference frames.

d) Because the mass of an object might be different in a non-inertial referenceframe

e) Because he wanted to exclude the possibility of force pairs that aren’tequal and opposite.


Problem 512.

problems/short-prove-circular-K3.tex

In a few lines prove Kepler’s third law for circular orbits around a planet orstar of mass M :

r3 = CT 2

and determine the constant C and then answer the following questions:

a) Jupiter has a mean radius of orbit around the sun equal to 5.2 times theradius of Earth’s orbit. How long does it take Jupiter to go around thesun (what is its orbital period or “year” TJ)?

b) Given the distance to the Moon of 3.84 × 108 meters and its (sidereal)orbital period of 27.3 days, find the mass of the Earth Me.

c) Using the mass you just evaluated and your knowledge of g on the surface,estimate the radius of the Earth Re.

Check your answers using google/wikipedia. Think for just one short momenthow much of the physics you have learned this semester is verified by the corre-spondance. Remember, I don’t want you to believe anything I am teaching youbecause of my authority as a teacher but because it works.


Problem 513.

problems/short-quarter-period-spring.tex

d

mk

A very simple model for a spring-driven toy gun is shown above, where a springwith spring constant k has been compressed a distance d from its unstretchedposition at the mouth of the barrel and a ball of mass m has been placed on a“massless” piston attached to the spring. At time t = 0 the gun is fired (thespring is released to act on the ball).

a) How long is the spring in contact with the ball? That is, how long afterthe trigger is pulled at t = 0 does it take for the bullet to emerge from thebarrel?

b) How fast is the bullet travelling as it emerges?


Problem 514.

problems/short-rail-gun.tex

(3 points) The U.S. Navy just last week demonstrated a magnetic rail gun firinga 10 kg projectile at Mach 7 (7 times the speed of sound). How much kineticenergy did the particle have?


Problem 515.

problems/short-range-of-stream.tex

0P 02P 0P

aP

a 2a a/2H

R Rba R c

A

Three pressurized water tanks are drawn up above, together with the differentrelative pressures on top of the water in the tank. The pressure outside of thetanks in all three cases is air pressure Pa, and P0 > Pa. All three tanks have thesame cross sectional area A at the top and the same height of fluid H , but arebeing drained through pipes of different cross-sectional areas as shown. Youmay assume that a ≪ A, and the ranges in the figures are obviously not drawnto scale.

Rank the relative range of the fluid emerging from the bottom pipe, assuminglaminar flow and no resistance in the pipe or tank (that is, list the three rangesRa, Rb, and Rc in ascending order, with <, > or = signs as needed).

Be careful, as the flow from the three pipes will (probably) not be the same!


Problem 516.

problems/short-rank-barrel-blowout.tex

A B C

In the figures above you see a side view of three containers filled or partiallyfilled with water and open to the air on the top, bottom, and all sides. Thecross sectional areas of the bottoms are all the same. You may assume thatthe containers are symmetric so that the cross sectional areas in the picture areindicative of the total amount of fluid in each container.

Rank the three figures by the order of the net force exerted on the bottom ofthe containers by the fluid inside, least to most (where equality is a possibleanswer) and indicate in a few words your reasoning and the basis of your answer.


Problem 517.

problems/short-rank-fluid-flow.tex

a) A = 30 cm1

2, v1 = 3 cm/sec, A 2 = 6 cm2

12, v1 2

2= 10 cm = 8 cm/sec, A = 5 cmb) A

, v12c) A 1 = 20 cm2 = 3 cm/sec, A 2 = 3 cm

In the figure above three different pipes are shown, with cross-sectional areasand flow speeds as shown. Rank the three diagrams a, b, and c in the order ofthe speed of the outgoing flow.


Problem 518.

problems/short-rank-sinking-masses-different-rho.tex

gm

DAk

ρ

A)

gm

DBk

2ρ

B)

In the figures above, two identical springs (with spring constant k) are attachedto rods above two identical containers filled with two different fluids withdensities ρ and 2ρ respectively. At the other end, the springs are attached toidentical lead blocks that would ordinarily sink into the fluids if the springwere not present. The distances DA and DB represent the total length of thestretched springs required to suspend the masses at equilibrium in the fluids.

Which spring do you expect to stretch the most (or will they both stretchidentical amounts)?


DA > DB DA < DB DA = DB


Problem 519.

problems/short-rank-the-effective-spring-constant.tex

k k k

k

k

k

k k

k

k

A

D

B

C

In the figure above you can see several combinations of identical springs. Rankthem in terms of their effective spring constant, in increasing order (whereequality is allowed). A possible answer is C < B = A < D.


Problem 520.

problems/short-rank-the-floating-fish.tex

A

B

C

D

Four fish are all floating at equilibrium at different depths, as shown. Rank thefish in order of increasing buoyant force. Assume that all four fish have thesame shape and differ only in size and depth. An acceptable answer might havethe form: C, D, B, A.


Problem 521.

problems/short-rank-the-orbits-2.tex

a

b

c

Sun

r

In the figure above three possible orbits of a planet of mass m around the sunare drawn. Rank them in the order of increasing angular momentum (andgive some brief argument or reason why you picked the order you picked).


Problem 522.

problems/short-rank-the-orbits.tex

a

b

c

Sun

r

In the figure above three possible orbits of a planet of mass m around the sunare drawn. All three orbits have the same distance of closest approach to thesun (r) as shown.

Rank the orbits a, b, c in the order of increasing total mechanical energy.


Problem 523.

problems/short-rank-three-flasks.tex

H

In the figure above three flasks are drawn that have the same (shaded) crosssectional area of the bottom. The depth of the water in all three flasks is H ,and so the pressure at the bottom in all three cases is the same. Explain howthe force exerted by the fluid on the circular bottom can be the same for allthree flasks when all three flasks contain different weights of water.


Problem 524.

problems/short-receding-moon.tex

The moon is actually slowly receding from the earth in a nearly circular orbit ata rate of around 4 centimeters per year. When T. Rex looked up at the sky, themoon was therefore 2400 or so kilometers closer to the earth than it is today.

Was the month (the period of the moon in its orbit around the earth) likely tobe longer or shorter than it is today, assuming only radial forces between theearth and the moon during all that time?


Problem 525.

problems/short-relative-velocity-freight-car.tex

0v

bv

m

A freight car is travelling at the constant speed v0 relative to the ground tothe right as shown. Inside the freight car, a block of mass m is given a push sothat it slides backwards at a constant speed vb relative to the freight car ona frictionless floor. The freight car is, of course, much more massive than theblock.

Circle the correct statement:

a) The speed of the block relative to the ground is v0 + vb.

b) The speed of the block relative to the ground is v0 − vb.

c) The speed of the freight car relative to the block is vb − v0.

d) The speed of the freight car relative to the block is vb + v0.

e) The speed of the block relative to the ground is vb.


Problem 526.

problems/short-resonant-string.tex

m

electrical oscillatorL

frictionless pulleyf

A massless hanger is suspended from a string with mass density µ attachedat one end to an electrical oscillator that vibrates at a fixed frequency f . Theother end is draped over a massless frictionless pulley. The length between thesetwo (approximately fixed) ends is L. You should recognize this apparatus asbeing very similar to one you used in a lab.

a) Sketch the envelope of the third harmonic around the string above.Label the nodes and antinodes. What is the wavelength of the thirdharmonic of this string?

b) Determine the mass m that needs to be hung on the hanger, as shownso that the frequency f is in resonance with the third harmonic of thestring in terms of the given or well-known quantities. (Note: The m ismuch greater than the weight of the hanging string.)


Problem 527.

problems/short-rolling-a-spool.tex

FF

F

(a) (b) (c)

In the figure above a spool wrapped with string is sitting on a rough table. Thecoefficient of static friction is large enough so that (for the given force ~F ) thespool will not slip on the surface. For each figure, indicate the direction thatthe spool will roll.


Problem 528.

problems/short-rolling-vs-sliding-ball-on-hills.tex

icy

rough

v

vblock

ball

v = 0 at t = 0

A)

A ball rolls down a rough incline without slipping. At the bottom of the inclinethe translational speed of the ball is:

a) greater than

b) less than

c) the same as

the speed of a block that falls straight down from the same initial height?

B)

It then slides up a frictionless hill on the other side. Is the height it reachesbefore its translational speed reaches zero:

a) higher than it started from

b) to the same height it started from

c) not as high as it started from

on the slippery hill?


Problem 529.

problems/short-scaling-of-wave-energy.tex

String one has mass µ and is at tension T and has a the travelling harmonicwave on it:

y1(x, t) = A sin(kx − ωt)

Identical string two has the superposition of two harmonic travelling waves onit:

y2(x, t) = A sin(kx − ωt) + 3A sin(kx + ωt)

If the energy density (total mechanical energy per unity length) is E1, what isE2 in terms of E1?


Problem 530.

problems/short-slipping-rod.tex


Problem 531.

problems/short-sound-barrier.tex

Fred is standing on the ground and Jane is blowing past him at a closest distanceof approach of a few meters at twice the speed of sound in air. Both Fred andJane are holding a loudspeaker that has been emitting sound at the frequency

f0 for some time.

a) Who hears the sound produced by the other person’s speaker as single fre-

quency sound when they are approaching one another and what frequencydo they hear?

b) What does the other person hear (when they hear anything at all)?

c) What frequenc(ies) do each of them hear after Jane has passed and isreceding into the distance?


Problem 532.

problems/short-sound-in-water-versus-air.tex

a b c

A

B

Sound waves travel faster in water than they do in air. Light waves travel fasterin air than they do in water. Based on this, which of the three paths picturedabove are more likely to minimize the time required for the

a) Sound

b) Light

produced by an underwater explosion to travel from the explosion at A to thepickup at B? Why (explain your answer)?


Problem 533.

problems/short-spinning-space-station.tex

Hub

People

Main ring

ω

The United Nations finally builds an international space station that is spun onits axis to create artificial “gravity”. The director (who is a political appointeewho never took physics) holds a press conference in an auditorium in the hub,and the entire population of the station climbs “up” a ladder from the mainring to attend. “This is a great achievement,” he announces, “a space stationthat spins at precisely one turn every twenty seconds, creating an accelerationof one gravity at the floor of the main ring.”

He is then interrupted by a polite cough from his aide, who excelled in Physics53 at Duke. “I’m sorry, sir, but that isn’t quite true,” she says. “Even if it was

nearly true this morning before this meeting, haven’t we all climbed up into thehub?”

State the principle of physics she used to arrive at this conclusion, and indicatewhether the angular velocity of the station’s rotation is likely to be largeror smaller than it was that morning, when all of the station’s workers andvisitors were far from the hub walking around on the floor of the main ring.


Problem 534.

problems/short-tides.tex

(4 points) It the midnight on the day of a full moon. Werewolves are outhowling, lovers are out appreciating the moonlight, fishermen are out fishing.Are the high tides of the day unusually high (spring tides) or unusually low(neap tides)? Draw a picture indicating the relative position of the Sun, theMoon, the Earth and the expected tidal bulge.


Problem 535.

problems/short-timed-trajectories-2.tex

If a mass m1 is dropped straight down from the top of the Duke Chapel fromrest, and another mass m2 > m1 is thrown sideways from the top of the Chapelat the exact same time (with no vertical component to its velocity):

a) Which one reaches the ground first?

b) Which one is going faster when it reaches the ground?

c) Does the answer (to either of these two questions) depend on the (relativesize of the) masses?


Problem 536.

problems/short-timing-lightning.tex

The lightning flashes, and you start counting. Five seconds later, the sound ofthunder arrives. Roughly how far away was the lightning strike? (In meters,please...)


Problem 537.

problems/short-torque-the-spinning-wheel.tex

d

a

b

c

F

F

F

F

Fc

F

Fa

Fbd

A rapidly spinning massive bicycle wheel similar to the one demonstrated inclass is drawn above. It is spinning in the direction shown (into the page at thetop). On the two handles four possible force couples are drawn, labelled a, b,c and d. Only one force couple, if applied, will rotate the axis of wheel so itsleft handle moves out of the page while the right handle is deflected intothe page (as shown). Circle the correct pair below:

a b c d

As always, it is good to say a few words about your reasoning.


Problem 538.

problems/short-two-orbits-2.tex

ba

R8R

Two identical satellites are in circular orbits around the earth, one at radius Rand the other at 8R.

a) Circle the satellite that has the longest period.

b) How much larger is its period? (Express the larger period in terms of theshorter period.)


Problem 539.

problems/short-weightless-orbit.tex

It is a horrible misconception that astronauts in orbit around the Earth areweightless, where weight (recall) is a measure of the actual gravitational forceexerted on an object. Suppose you are in a space shuttle orbiting the Earthat a distance of two times the Earth’s radius away from its center (a bit morethan 6000 kilometers above the Earth’s surface). What is your weight relative

to your weight on the Earth’s surface? Does your weight depend on whether ornot you are moving? Why do you feel weightless inside an orbiting shuttle? Canyou feel as “weightless” as an astronaut on the space shuttle (however briefly)in your own dorm room? How?


Problem 540.

problems/short-which-mass-accelerates-most.tex

m1 m2

In the figure above, two masses m1 < m2 are connected by a stretched (massless)spring and released from rest. There is no friction or gravity acting. Answerthe following questions:

a) Which mass experiences a larger force (or are they the same)?

b) Which mass experiences a larger acceleration (or are they the same)?

c) After 3 seconds the momentum of mass m1 is ~p1. What is the momentumof mass m2 at that instant?


Problem 541.

problems/short-windshield-and-bug-ML.tex

Sometimes you’re the windshield, sometimes you’re the bug. Today you are thebug. A cement truck with a mass of 25 metric tons collides with you at a speedof 50 meters/second while you hover above the road. A number of things flashthrough your buglike mind right before the impact, some of them true and wiseand some of them false and foolish. Circle the true and wise statements in thelist below:

a) Hah! At least I’m going to exert as much force on the truck as it is on meduring the impact!

b) If I recoil off of its windshield elastically, I’ll emerge unscathed movingdown the road at twice the speed of the truck!

c) Darn! The truck is going to change my momentum by a huge amount andI’m hardly going to alter its momentum at all!

d) If I splatter and stick to his windshield, at least our mutual kinetic energywill be conserved!

e) My one regret is that I didn’t take physics at the Duke Marine Lab thissummer. If I had, I wouldn’t be out here hovering in front of onrushingcement trucks!

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